Let's go ahead and get these out of the way.
Burrell versus Damon Act I
Whether you want Johnny Damon (or Milton Bradley or Jim Thome) or not, just because someone does want one of those players does not mean they have zero confidence in a Pat Burrell bounce back. There is reason to believe Burrell will play more like his old self than the old-looking self of last year. There is also reason to believe that Damon will outperform Burrell as a full-time DH. It's completely possible to think Burrell is capable of being an above average hitter again while also wanting Damon to replace him.
Burrell versus Damon Act II
CHONE projects Burrell at a .329 wOBA while Marcel says .346. Damon at .352 and .353. The difference between a .352 wOBA and a .346 wOBA is marginal and indifferent. The difference between a .330 wOBA and a .352 wOBA over 600 plate appearances is more than a win. Important for a team that needs every win it can get. The difference could be as little as a run or as wide as a win. The Rays have access to information that we'll never see, and if they're barking up the best free agent DH trees, then maybe they just think Damon is a safer bet.
Burrell versus Damon Act III
Burrell is getting $9M next season whether he plays for the Rays, Jays, or sits on the bench munching on ham with honey glaze. If his salary could've been moved without the Rays eating some, then it probably would have happened. The best scenario here is something like eating 75% of Burrell's deal while getting a random pen arm and then signing Damon with enough deferred money that his deal really isn't worth $6M or whatever. Yeah, that's a lot of money to invest the DH spot, more than we should ever hope for, but the Rays understand the concept of sunk costs and just because they have Burrell doesn't mean they won't spend the money to upgrade if they feel it's the right move.
It's really not clear cut. Keep that in mind.