Properties of language that can be enumerated in a finite amount of time with an enumerator?
If the language can be enumerated in finite time, then the language itself is finite. Almost all natural classes of languages include all finite languages: they're regular, context-free, decidable, recursively enumerable, co-RE, ...

I'm not sure what else you'd want to say about finite languages compared to RE languages. It's enough to say that the language is finite; for any property you might be interested in, it's usually very straightforward to prove that finite languages do or do not have that property.