Actually, the two things have very similar meaning that studying Korean is fun. But in no.1, you can see the noun (명사) was used to make it subject in the sentence. and as for no.2 , if it is "공부하는 것은 재미있어요", then, you can see the form of 'V-ing' was used (공부하다 + -는 것) to make it subject! So, consequently, there is no big difference between them in meaning, but a little bit difference in respect of how they were used, I think! i hope it helps!

2. 한국어 공부하는 '거' 재미있어요 is correct both sentences make sense and pretty much the same in terms of meaning and nuance