Well, you should be able to upload and run any classes you want, providing that JME has enough of the basic dynamic classhandling facilities we know and love from J2SE (I'd have to RTFM). And as long as you don't run afoul of the sandbox.
However, web-based has a couple of advantages. First, however, one big
disadvantage: you have to have a live Internet connection to run webapps!
Some advantages are:
1. Software updating. You only have to update the servers, not every phone.
2. Wider hardware support. As mentioned, many smartphones have web browsers. But my phone came with Windows Mobile, and you know how many Windows products come with
Java built-in.
3. Server security. Data can be kept more secure in a central location. And by "secure", I mean backed-up as much as I mean "keep your nasty little hacking paws off my data!".
4. Maintainability. It's a lot easier to find people who can maintain stock webapp code than the more specialized JME platform and you have a wider variety of proven platforms to build on.
Of course, there are certain things that really shouldn't be held hostage to an Internet connection. For these, a blended JME/webapp approach is worth considering.