It gives the player (who often is out of the rotation) a chance to get picked up by a team that wants them. The difference in guaranteed amount that they give up would likely be less than or equal to what they can get from a new team. . . with the opportunity to get quite a bit more or multiple years on the open market instead.
That's why agents discourage teams from making waiver claims. The benefit comes from making more money or having the opportunity to pick your team. Any team with cap space can circumvent that by making the waiver claim for the original contract. Jazz have that cap space and can easily get by without Shelvin Mack (or Jeff Withey) to grab Andrew Bogut if they want him and keep him off the market - and away from the Cavs, Celtics, Rockets, Warriors or Spurs.
That doesnt answer the question: Why would any player agree to a buyout if the process is the same as being cut/waived?