What is the difference between nauczy się vs uczy się?
We have in all Slavic languages IMPERFECTIVE and PERFECTIVE verbs.
The imnperfective verbs show the process of something - uszy sie
the perfective verds show the result of the action - nauszy sie
Sorry, I don’t have polish letters
The difference is that one is perfective and the other one is imperfective verb as evgueny40 noticed.
“Uczy się” - means someone “is learning”
Whereas:
“Nauczy się” - means someone “will learn something”
Examples:
“On UCZY SIĘ na sprawdzian z matematyki.” - He is learning/learns (both are correct) for math test.
“Z pewnością bardzo szybko NAUCZYSZ SIĘ polskiego.” - You will surely LEARN Polish very fast.
If you want more examples or have another questions, just let me know 
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over one year later but thanks for the answer LOL