I would have preferred Chris Andersen over Collins. I assume both would have been available for the minimum.
Birdman: 11.9% Offensive Rebound Rate; 23.3% Defensive Rebound Rate; 17.8% Total Rebound Rate
Collins: 4.6% Offensive Rebound Rate; 13.6% Defensive Rebound Rate; 9.1% Total Rebound Rate
Compared to last year's Celtics team, Andersen would have been our best rebounder by a fairly large margin (KG's TRB% was 16.8%). Andersen would have been our best offensive rebounder, as well, and would have been second only to KG in defensive boards. Compared to Collins, Andersen is literally twice the rebounder that he is. Collins' rebound percentage would have been 9th on the Celtics, virtually tied with Paul Pierce.
In other words, Andersen is an excellent rebounder, and Collins is very, very bad.
Birdman: 12.4 points per 36 minutes; 0.4 assists per 36 minutes; 1.3 turnovers per 36 minutes; 54.6% FG%; 61.0% FT%; Offensive Rating: 119 (!!)
Collins: 4.6 points per 36 minutes; 1.2 assists per 36 minutes; 1.2 turnovers per 36 minutes; 40.0% FG%; 46.7% FT%; Offensive Rating: 87 (!!)
There's no comparison on this end of the floor. Andersen is a much better scorer, and a much more efficient scorer. Collins does average more assists, but his turnover rate (rather than raw turnovers) is much worse: 17.7% vs. 10.6%.
Birdman: 3.4 blocks per 36 minutes; 1.4 steals per 36 minutes; Defensive Rating: 99
Collins: 0.4 blocks per 36 minutes; 0.5 steals per 36 minutes; Defensive Rating: 104
Birdman: 0.74 points allowed per possession (40th overall); 0.79 points allowed per possession allowed on post-ups (103rd overall)
Collins: 0.78 points allowed per possession (75th overall); 0.84 points allowed per possession allowed on post-ups (143rd overall)
Statistically, Birdman's defense was better than Collins.
Andersen is known to be a great teammate. He's a high-energy player and a crowd favorite. On the negative side, he's been injured for parts of the last two seasons, and he was the subject of a criminal investigation involving photographs of an underage girl (while his lawyer says he was the victim of an extortion attempt by that girl -- who claimed she was 21 -- and her mother.) Andersen also has a history of hardcore drug use, but seems to have kicked the habit.
Collins is also known as a great teammate, and he willingly fills his role. He's more likely to rack up DNP-CDs than he is to miss games due to injuries.
It's hard to argue that Collins is in the same class as Andersen as a player. Collins is healthier and has fewer question marks, but he's also an inferior player.