The Aztecs will have to heavily restructure their entire way of warfare if they ever hope to become an international power since Aztec combat was focused on taking captives for sacrifices. That mindset will have to change to “aim for the kill” if they hope to survive further contact with Europe.
As for the weapons, it will be virtually impossible to reverse engineer gunpowder when they have no idea what’s actually made of or how it’s manufactured (though if someone from the expedition has some basic knowledge that could speed things up) .
You can easily go around this by having the majority of Columbus’s men defect to the Aztecs , Columbus was notorious for being an unpleasant person and cruel even for the standards of his time, its not implausible for him to make a mistake or two that cost him the loyalty of his crews and with a generous bribe from the Aztecs it could be easy to sway them. If there are blacksmiths amongst them then they should know how to make steel or at least have a basic idea for it.
Edit : as pointed out this can also backfire horribly on the Aztecs if the crew is instead taken in by any of the people subjected by the Aztec empire, and use the technology to gain independence . In that case either the Aztecs will have to change to match them or more likely be forced to accept them as equals . Best case scenario is that these newly independent civs remain neutral. I can even see them forming a proto-federation though wether or not the Aztecs are part of it I cannot be sure (assuming it’s not an alliance against them).
Something I also forgot was smallpox. If smallpox spreads I suspect it will actually be less devestating somewhat since the Americans won’t be dealing with an invasion and might be able to deal with it more or less effectively or at least control so its not an apocalypse level plague. By the time of re-contact the people of Mesoamerica might develop partial resistance to smallpox.