We all know how much fun it is to play other people's music. In fact, whether you play jazz, rock, pop, or classical music, that's what we do most of.
There are only 2 ways to cover a song, and they are both valid.
The first way is to play it basically the same as the original artist did. This is what The Rolling Stones did Mississippi Fred McDowell's blues tune "You Gotta Move."
Listen to how similar the 2 recordings are:
You might be thinking, "Why would anyone record a song in the same manner as the original rendition?" Well, it's simply because we all enjoy playing covers at home, and sometimes artists just record songs they way they like to play them. In Rolling Stones' case, they began their career as a cover band and knew a great many of these traditional blues tunes. This is the way they liked to play "You Gotta Move." They probably also wanted to put something slow and bluesy on their "Sticky Fingers" album, and this fit the bill perfectly. In any case, their musical identity comes shining through.
The other choice is change the approach to the song. Play it faster, slower, with a different orchestration or musical style. Here's what Eric Clapton and "Cream" did with another old blues chestnut:
Same song; very different approach.
Since you already know how to play songs the original way, why don't you have some fun and experiment a little? Take a few songs you know very well and transform them into something different. Here's a little more inspiration to check out. Have fun!
Take your left hand playing to a new level with my free ebook: Left Hand Techniques for Jazz Piano
You'll also get my weekly jazz newsletter with practice tips and inspiration