Both words can mean close by in position, quality, and accuracy.
The OED's examples for "proximate" illustrate its sense of closeness in the meaning of words, geographical position, and strength of commercial relationships. The substitution of "approximate" would be apt in these examples, given its definition "Very near, in position or in character."
From History of the Berwickshire Naturalists' Club, Volume 1 (which is quoted in part as an example from the OED) describing the suckers of an octopus:
[T]oward the base of the arm they are larger and not in contact, but
they soon become approximate, and gradually lessening in size they
become very minute at the tips....
"Proximate," however has a usage that its cousin lacks -- next or previous in order. This may be temporal (as in the "proximate future," something that will happen next as opposed to remote events to come), or causal
The jury had to determine the proximate cause of the collision
or with respect to process, particularly in chemistry, in which "proximate analysis" of a compound means determining the compound's precursors.
Both words have synonymous adverbial forms -- "proximately" and "approximately."
Both words have synonymous noun forms, but with different morphemes -- "proximity," "proximateness" and "approximation."
Only "approximate" is a verb.