While I admit firstly I'm not an American and can't claim to be an expert, I'd point out that one of the strengths of U.S. states having their own local regulations is that each can adopt laws and regulations as they see fit and see which works best (or at least, best for them).

In terms of gun licensing, in the U.S. it's not politically clear (i.e. there is no consensus at the federal level) which state has the best policy on gun ownership (in fact it's one of the most volatile, ongoing political and social debates in the U.S. of course), so blanket legislation that states must commit to abide by the gun possession regulations of other states, when their conditions may vary wildly, seems both unlikely and unwise, not to mention extremely unpopular in certain corners.

You can compare the rationale behind states recognizing each other's driver's licenses, but the personal criteria for licensing does not vary nearly as greatly as it does for guns, and certainly not as contentiously.