Player Versions are meant to emulate real football jerseys used on the football pitch, hence why the player versions tend to be more slim-fit than normal t-shirts. If you're somebody who likes slim fitted clothing or something that you can use while playing sports, than the player version is perfect for you.
As you can see in the the picture, Fan Versions tend to be more normal sized than Player Versions. These are meant to emulate everyday clothing that you can use for your fashion needs. So if you're looking for something that's gonna stun your friends while with them or something more loosely fitted, than the Fan Versions is for you.