For Huck, being on the river means freedom from his pap, from school, from church, from dressing nice, from sitting up straight, from not swearing, etc. He's able to use this adventure as a method for escaping from all of the people trying to tell him how to live his life - the civilized version that Miss Watson wants him to have; and the uneducated version that Pap wants him to have.
For Jim, however, his freedom on the river is exactly that, freedom. This is his opportunity to get out of slavery for good. After finding out he was going to be sold (and getting blamed for Huck's death), he takes to river becoming an escaped slave. For Jim, the adventure is hoping that at the end of this he will be a free man with the opportunity to buy his wife and kids back again.