What is the meaning of uppajjati versus upapajjati? In the Pali there are two terms, which are (generally) used contextually in two different ways:


uppajjati (and its past particle uppanna)
upapajjati (and its past particle upapanna)


What do these terms actually mean and what is the distinction between them? Importantly, how are these terms constructed or derived in Pali to differentiate their meaning? 

Of note: the PTS dictionary makes the rather vague comment:


  doubtful whether a legitimate form as upa + pad or a diaeretic form of uppajjati = ud + pad. In this case all passages ought to go under
  the latter. Trenckner however (Notes 77) defends upa˚ &amp; considers in
  many cases upp˚ a substitution for upa. The diaeresis may be due to
  metre, as nearly all forms are found in poetry. The variant reading
  upp˚ is apparently frequent; but it is almost impossible to
  distinguish between upap˚ and upp˚ in the Sinhalese writing, and
  either the scribe or the reader may mistake one for the other
I think there are two rather different words confused here.

One is Sanskrit "utpada"/"anutpada" (to arise / not to arise) and its Pali corruption "uppada" with a past form "uppanna".

Another one is Sanskrit "upaprajati" (something like "offshoot" or "derivative" - meaning something that is born off a family but separate) and the Pali corruption of its past form, "upapanna".

Google dictinary for "offshoot" gives the following meanings:


  
  a thing that originated or developed from something else.
  "commercial offshoots of universities"
    synonyms: outcome, result, (side) effect, corollary, consequence, upshot, product, by-product, spin-off, development, outgrowth, fallout
    "rap music began as an underground offshoot of disco"
  subsidiary, branch, adjunct, derivative
    "the company now controls several offshoots"
  


Checking the examples of sutta usages in @ChrisW's answer, I think they all make sense.