Two very different examples:

Modest Mussorsky’s “Pictures at an exhibition” was originally written for piano (that was all the poor guy had available), and it took Maurice Ravel to turn it into an orchestral version and make it popular world wide. Both versions are great and have their own qualities, but I far prefer Ravel’s version (Although Khatia Buniatishvili’s rendition here is so really really good).

Bob Dylan’s “Lay Lady Lay”. The original is… well… it’s a Bob Dylan song. There is a really good cover version by Ministry that, althought technically is the same song, takes it to a whole other level. Love it.