These are two different things.
You should not conflate them.
How would you describe someone that continues to perform and action solely because they have observed someone else performing that same action, but do not know the reason.
The colloquial expression that comes to mind here is "monkey see, monkey do."
Or, they perform an action in a certain way and do not change to a better method, only because they believe the current method must be the best way to do it because it has been done that way previously.
You could say they are set in their ways or hidebound .
english.stackexchange.com/users/2303/Robusto
The term Cargo Cult could fit this situation.
See the Wikipedia page http://en.wikipedia.org/wiki/Cargo_cult and scroll down to 'Other uses of the term'.
english.stackexchange.com/users/2819/chimp
At one extreme, sheeple can be used to describe people who follow the masses and act without thinking for themselves.
english.stackexchange.com/users/2988/Brian Nixon
In a commercial environment, especially at a large company (although small ones are by no means immune to this effect), one might say that such people are suffering from "institutional inertia".
english.stackexchange.com/users/2490/Hellion
they perform an action in a certain way and do not change to a better method, only because they believe the current method must be the best way to do it because it has been done that way previously.
Traditionalist.
english.stackexchange.com/users/3461/msh210
One could say they are "set in their ways".
But that's more the latter scenario.
For the first... well, sheeple (which is a portmanteau of "sheep" and "people", btw) doesn't quite cut the mustard here, but it's not far from it, as it is "doing something without thinking".
english.stackexchange.com/users/2701/Jürgen A. Erhard
There's a slightly obscure word that means something very close to this: perseverate.
The word comes from the medical context (discussing mental illness), and means to continue doing something after the reason has disappeared.
english.stackexchange.com/users/1812/Henry
Slightly different shade of meaning: do it by rote ,
that is, you've learnt it by repetition and do it without knowing why.
english.stackexchange.com/users/5130/Concrete Gannet
Some previously-mentioned words (eg hidebound , traditionalist , perseverate ) seem good.
Reactionary
(“Opposed to change; urging a return to a previous state” or “Very conservative”), conservative (“A person who favors maintenance of the status quo or reversion to some earlier status”), and stick in the mud (""one who is slow, old-fashioned, or unprogressive; an old fogey) must also be mentioned.
Stick in the mud dates from the late 1500's, according to phrases.org.uk , which also says figurative phrase 'stick in the mud' ... was preceded in the language by earlier versions, for example 'stick in the briers, clay, mire' etc.
These were usually applied to people who remained in a difficult situation, either by choice or because they were stuck.
english.stackexchange.com/users/11433/James Waldby - jwpat7
How would you describe someone that continues to perform and action solely because they have observed someone else performing that same action, but do not know the reason. Or, they perform an action in a certain way and do not change to a better method, only because they believe the current method must be the best way to do it because it has been done that way previously.  What word/phrase/expression would you use to describe this scenario?
Continuing to do something just because it was done before, without knowing why
english.stackexchange.com
<word-choice><phrases>
english.stackexchange.com/questions/8901
english.stackexchange.com/users/3675/Daniel
These two concepts are not the same thing and should not be conflated. A potential colloquial phrase to describe one of these scenarios would be "monkey see, monkey do". Additionally, you could say that such people are "set in their ways" or are "traditionalist". The term "Cargo Cult" may also refer to this phenomenon. The question seems to raise a search for several terms and some believe they are not the same. Some expressions suggested are "monkey see, monkey do," or to be "set in one's ways/hidebound". The term Cargo Cult was also suggested with a Wikipedia reference. Alternatively, "sheeple" was suggested, although some disagreed with this suggestion. "Persevate" was also another term proposed.
These two concepts are not the same thing and should not be conflated. A potential colloquial phrase to describe one of these scenarios would be "monkey see, monkey do". Additionally, you could say that such people are "set in their ways" or are "traditionalist". The term "Cargo Cult" may also refer to this phenomenon. Several expressions can be used to describe such people, including set in their ways; traditionalists; hidebound; suffering from institutional inertia; sheeple or being part of a Cargo Cult.
Suggested terms and phrases include set in their ways; hidebound; cargo cult; sheeple; traditionalists; and perseverate. There is a wide range of terms that might describe such people, including "hidebound", "sheeple", traditionalist" or "perseverate".
There are two different meanings at play. Common expressions like "monkey see, monkey do" Set in ways/hidebound "Cargo Cult" could be the meaning sought.  "Sheeple" is a term for people who follow the masses blindly. Someone "set in their ways" "Sheeple" not appropriate Persevarate  Other synonyms suggested
Several expressions can be used to describe such people, including set in their ways; traditionalists; hidebound; suffering from institutional inertia; sheeple or being part of a Cargo Cult.
There is a wide range of terms that might describe such people, including "hidebound", "sheeple", traditionalist" or "perseverate".