First, the whole .NET thing (Or rather, the lack of cross-platform compatibility). Yes, Mono is pretty good, but nowhere near perfect. (Although note that this is changing).
And second, enums are hideous in C#. One of the few things Java does better than C#.
I find it really annoying that C# enums cannot have constructors and variables.
You can do exactly what java calls enums with classes in C#, which in my view are better suited to having constructors and fields. But I agree it's all preferences.
No you cannot. You lose the single biggest benefit of enumerations - namely compile-time checking. (For instance: warning when a switch statement does not cover all cases.)
Try to do this in C#, and you'll see what I mean: http://snipplr.com/view/42422/the-planet-enum-example/. Either you end up with it being ~3x as verbose (if not more), or you lose the compile-time benefits, or both.