infantrycak
Hall of Fame
It depends on whether the player is a vested veteran or not but it's effectively the same thing for the team getting rid of the player.
For the player, it's a very different thing. If you're a vested veteran and you're cut, you're free to sign with whoever you want. If you're not a vested veteran and you're cut, teams can place a claim on you. The teams get priority based on their previous year's record. So a player gets cut and a 2-14 team and a 14-2 team places a claim on him, the 2-14 team gets the player.
To add on to what TPN said - a cut player negotiates a new contract with any new team. A team taking a player off waivers is taking their old contract as well.