They are synonyms; either one can work. It is my feeling that inside the LDS Church culture, proselyte is more common. However, that’s definitely not the case in general. I used Mark Davies’s Corpus of Contemporary American English (COCA) and found that it includes 403 instances of some form of the word proselytize used as a verb, but only 1 instance of some form of proselyte used as a verb. (The search was proselyt*.[v*] for anyone who’s interested.)
Using Mark Davies’s corpus of [LDS General Conference] 2 talks seems to confirm my hunch: I found 3 instances of some form of proselytize, while there were 158 instances of a verb form of proselyte. (Note that the raw numbers can’t be compared across corpora, but the ratios within each corpus are very telling.)
— Laser Jock, in response to [my question] 3 about proselytize vs. proselyte.
So, I stand [somewhat corrected] 4. However, even though “proselyte” is a real word, it can still be added to the list of uncommon words/phrases that for some reason are quite common amongst Mormons.