Q:

"endure" vs "perdure" vs "persist"

All definitions via The Free Dictionary
perdure means 

To last permanently; endure

endure is

To continue in existence; last

persist means

To be obstinately repetitious, insistent, or tenacious.
To hold firmly and steadfastly to a purpose, state, or undertaking despite obstacles, warnings, or setbacks.
To continue in existence; last

UPDATE  
It appears the answers and comments have explained the difference between persist and perdure 

To persist is to remain wholly over time. To perdure is to remain over time despite changes. Objects persist and events perdure.
  @Philosopher

but the difference between “perdure” and “endure” remains obscure.  It seems as though both terms are extremely similar in meaning. Is the difference technical? Could someone please explain the difference in meaning and in usage between perdure and endure? 

A:

The odd one out, and by far the rarest of the three, perdure has more “throughness” to it, implying that something has not just endured, but in fact persisted through or despite something else. It has more of a survival connotation than the others. Selected OED citations:

1854 Hickok Mental Philos. 76 ― The mind perdures while its energizing may construct a thousand lines.
1973 Boilès & Horcasitas tr. M. León-Portilla’s Time & Reality in Thought of Maya ii. 33  ― For longer than a millennium and a half, not a little of Maya symbolism has perdured.
1979 Nature 22 Mar. 348/1  ― Thus enough maternal gene products (mRNAs or proteins) may perdure in embryonic cells to allow normal segmentation and cuticular differentiation. 

And for the resulting perduring:

1890 J. Skinner Dissert. Metaphysics 109 ― The Soul is revealed intuitively as a perduring living agent or entity.
1977 Dædalus Summer 63  ― The assignment and reassignment of meaning must be investigated as processes in the domain of resilience possessed by each population recognizing itself to be culturally perduring.

A:

The difference between perdure and endure is that perdure has a connotation of continuing to last forever, or at least until death, while endure has a connotation of only continuing to last until some specified time.
For example, in this sentence:

"His plays have endured for more than three centuries."

You wouldn't use perdured here because it is not guaranteed that his plays will continue to last. You also wouldn't use persist because there is not a connotation of overcoming an obstacle in this particular sentence. A way to modify to better fit persist: "His plays have persisted for more than three centuries, despite not being relatable to a modern audience".
The user should be wary of using perdure in the past tense, but it is possible, as long as the user is absolutely sure that the subject will continue to exist. For example, "his love for her has perdured for fifty years" is acceptable, although personally I would use endure in this case despite the negative connotation. However, for a future tense, i.e. "his love for her will perdure", perdure is the clear winner over endure, and persist is dead last for both tenses, and should never be used for matters of love because it has the most negative connotation out of all three options, suggesting that continuing to exist is somehow a hassle, i.e. "our love persists on" (as if the love is somehow stubborn. I wouldn't connote love with stubbornness).

Another example where it wouldn't make much sense to use endure:

"May you perdure faithful to the end", i.e. may you continue [in being] faithful to the end.

It's possible to use endure here but it sounds awkward, because faithful has a positive connotation but endure has a negative connotation, i.e. why would you suffer by being faithful? Also, endure to the end seems to suggest that the faithfulness is only enduring with the common objective of reaching an end state, but perdure to the end is suggestive of flourishing and living in the present, with little regard for the end. In other words, endure to the end already seems resigned to the end, which is pessimistic for such a sentence which is intended to be inspirational and life-affirming.
Using persist would work, but for different reasons; persist has the extra connotation of continuing despite something, so it is up to the user's discretion which connotation he wants to use. For instance, one would use persist if the subject's faithfulness is facing some obstacle such as the temptation to commit adultery. Perdure is the best option here because it has the most positive connotation and it suggests that his faithfulness is eternal, which gives you more bonus points in the romance department as well.

The word persist, when not used in contexts of overcoming struggle, is best used when whatever is continuing will eventually stop. For example, in the sentence:

"The rain persisted."

endured will not work because it is unclear what the rain is enduring: time?. Perdured doesn't work either because the rain will eventually stop. In another example, "The disease persisted" similarly endure doesn't work because, unless you're describing how the disease is fighting off your immune system, it's not clear what the disease is enduring. And once again, a disease cannot perdure because eventually it will either go away, or the victim will die. In both of these example sentences, there is a connotation of stubbornness; both the rain and the disease are being stubborn in their refusal to go away (but this does not mean that they will eventually not go away, and for this reason persist is the best choice for temporal states that last a long time, but eventually do go away).

In summary: use perdure in positive contexts indicative of eternal perpetuity,
use endure in negative contexts (of struggle), or contexts indicative of not being sure how long it will last, but it will continue to last despite that lack of knowledge,
and use persist in contexts of continuing in triumph and overcoming the obstacle, despite possibly the odds not being in your favor.
In short, perdure for perpetuity, endure for suffering through, and persist for overcoming the odds.
In terms of the length of time that the subject is continuing to exist: perdure suggests eternally continuing, endure suggests for an indeterminate time, and persist suggests for however long it will take to overcome the obstacle. However, use the above contexts along with this rule of thumb, because the emotional connotation is important as well.
Some more connotations:
~ perdure (confidence, certainty, the most long-term, the future) (whatever is perduring will continue to perdure [not necessarily related to the past]),
~ endure (unpredictable, ambiguous, indeterminate duration, the past) (whatever has endured may or may not continue to endure [says nothing about the future, or necessarily the present either]),
~ persist (stubborn, resolute, the most short-term, the present) (whatever has persisted is continuing to persist [but says nothing about the future]).
Of course, that's not to say that one cannot use perdure in a past tense, or endure in the future tense, or persist in the future tense, but I'm just speaking in terms of which word is most useful for which tense. For me, the following sentences sound awkward:
~"He perdured during his disease-ridden years" (it's kind of self-evident that the statement implies that he survived just through the dependent clause 'during his disease-ridden years', so the word perdured is redundant. It would be like saying "He survived during his disease-ridden years". Endured is a slightly better fit, but still suffers from redundancy. The best fit is persisted because it implies that he successfully overcame his disease and is now better for it. Endured merely says that he barely survived it, and says nothing about the future of his health either.)
~"His love will endure" (the problem with the future-tense of the intransitive endure is that it's ambiguous as to why the love has to endure anything at all, and it's ambiguous as to how long it will endure. In short, it lacks a sense of confidence in the future. It also sounds cliché in the way that vague statements usually do. Persist suffers the same problem here. Perdure is the best option because it automatically presupposes and confirms that his love will last, at the very least, until his death.)
~"The rain will persist" (how can we be sure that it will persist? That's impossible to say really. You can say that the rain persisted. You can say that the rain is persisting, but to say that it will persist sounds awkward, and it's better to modify this sentence with a quantifier, such as "The rain will persist for the next two hours". For this reason, persist is not a useful word for describing events that are ambiguous as to how long they will last. It is most useful in the present and simple past tenses, such as "The rain is persisting" or "The rain persisted".)

Definitions:
perdure
American Heritage dictionary

~[no object] To last permanently

Oxford U.S. English dictionary

~[no object] Remain in existence throughout a substantial period of time

dictionary.com

~[no object] to continue or last permanently

etymonline.com etymology (for perdurable)

mid-13c., from Old French pardurable "eternal, everlasting, perpetual" (12c.), from Late Latin perdurabilis, from perdurare, from per- "through, throughout; thoroughly; entirely, utterly" (intensive prefix) + durare "to endure"

~
Examples in Literature: finedictionary.com (under Usage)

endure
American Heritage dictionary

~[with object]

To carry on through, despite hardships; undergo or suffer

To put up with; tolerate

[without object]

To continue in existence; last

To suffer patiently without yielding

Oxford English dictionary

~[with object] Suffer (something painful or difficult) patiently
[no object] Remain in existence; last

dictionary.com

~1. [with object] to hold out against; sustain without impairment or yielding; undergo

[with object] to bear without resistance or with patience; tolerate

[no object] to continue to exist; last

[no object] to support adverse force or influence of any kind; suffer without yielding; suffer patiently

[no object] to have or gain continued or lasting acknowledgment or recognition, as of worth, merit or greatness

etymonline.com etymology

late 14c., from Old French endurer (12c.) "make hard, harden; bear, tolerate; keep up, maintain", from Latin indurare "make hard", in Late Latin "harden (the heart) against", from in- + durare "to harden", from durus "hard", from PIE *dru-ro-, from root deru- "be firm, solid, steadfast"

~
Examples in Literature: finedictionary.com (under Usage)

persist
American Heritage dictionary

~[no object]

To be obstinately repetitious, insistent, or tenacious.

To hold firmly and steadfastly to a purpose, state, or undertaking despite obstacles, warnings, or setbacks.

To continue in existence; last

Oxford English dictionary

~[no object]
Continue firmly or obstinately in an opinion or a course of action in spite of difficulty, opposition, or failure
Continue to exist; be prolonged

dictionary.com

~[no object]

to continue steadfastly or firmly in some state, purpose, course of action, or the like, especially in spite of opposition, remonstrance, etc.

to last or endure tenaciously

etymonline.com etymology

1530s from Middle French persister (14c.), from Latin persistere "abide, continue steadfastly", from per- "thoroughly" + sistere "come to stand, cause to stand still"

~
Examples in Literature: finedictionary.com
(under Usage)

As you can see, the dictionaries confirm that perdure should be used when the existence of the subject is permanent in nature, or desired to be permanent. Also, remember that endure can be used with objects, while the other variants cannot.
The etymologies show that perdure itself came later after the adjective perdurable (mid 13th century) (perdurable itself is the much more popular word), which itself came earlier than endure (late 14th century). The prefix per- (thoroughly, entirely, utterly) as an intensive is much stronger than the prefix en-, which further shows how perdure is not only connotative, but denotative of a larger length of time than endure.
However, this per- doesn't translate to persist as well because the latter part of the word, from sistere, means to come to stand still, or cause to stand still. And stillness is not connotative of duration/eternity as much as -dure from duro is.
This ending of sistere is present in words such as  exist and subsist, which also share the connotation of presence just like persist does.
Parts of Speech:
perdure: perdurance (n), perdurable/perdurant (adj), perdurably/perdurantly (adv-not standard), (I'm not including perduration because, like endurance, it can mean the capacity to endure, while perdurance is the action of perduring. Also, do not confuse perdurable for perdurant, as the former can also mean very durable.)
endure: enduringness (n), enduring (adj), enduringly (adv),   (I'm not including endurance because it has come to mean the capacity to endure, and not the act of enduring itself. Similarly, do not confuse endurable/endurant for enduring.)
persist: persistence/persistency (n), persistent (adj), persistingly/persistently (adv)

If I had to come up with sentences that best characterized the connotations of these words for the different tenses, they would go something like this:
Past
His love perdured. (and will continue to)
The cities endured. (says nothing about the future)
Hostilities persisted. (but will eventually stop)
Present
The influence of the Roman Empire perdures. (and likely will continue to do so for centuries to come)
The millennial tree endures. (but uncertain for how much longer)
The symptoms persist. (says nothing about the future)
Future
My legacy will perdure. (it will reverberate and forever have eternal effects [hyperbole])
My family's heritage will endure. (may be decades, centuries, but not forever)
His tenacity will persist. (not necessarily for his entire life)
~
As you can see, perdure is the most figurative and evocative out of the three because--paradoxically--its insistence on the certainty of the future ultimately gives more weight to its impact on the present. A love that perdures for decades can only accomplish that by being strong in the present. Meanwhile, persist is the most literal and faithful to its denotation. Endure is the most ambiguous of the three and perhaps for this reason it is the most versatile.

A:

In philosophy the distinction is important. To persist is to remain wholly over time.  To perdure is to remain over time despite changes.  Objects persist and events perdure.
A table persists over time.  It is, by and large, wholly present over time.
A birthday party perdures.  People come and go, the activities change, the tone changes, but the party continues over time.  It perdures. 

