the American ones are better, for the most part, off the shelf.
The Japanese Jags (generalizing big time here, since I've only played 3 or 4 of these) play very well and sound pretty good stock. But the pickups, I've found, are very brittle and thin-sounding. The electronics are also pretty bad compared to MIAs. The MIJ bodies are sometimes made of basswood, which isn't "authentic". Especially the '90s Jags, of which I've owned two (both basswood). I think basswood guitars can sound fine, but it's just something to consider.
The MIAs on the other hand, may not necessarily play a whole hell of a lot better, but the stock electronics are a step up, and some of the other details are there (like the aforementioned nice cases). The bodies are all made of alder and the paint has that Vintage Reissue style nitro finish. The pickups are noticeably less harsh. I own an AVRI Jazzmaster, but no Jag at the moment, so what do I know. If I was looking for a Jag now, I'd try to snag a cheap AVRI, but the MIJs are nothing to sneeze at. And they can be upgraded.