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There are the surface reasons of why he would not sign: he's been with the Cavs his whole career, LeBron gives him a better chance to win a championship than the Hawks, he knows everyone on the Cavs, etc. I think there might be a deeper, and more troubling, reason why Ilgauskas would not sign with the Hawks: the example of Joe Smith.
Joe Smith drew the interest of several contenders this off-season to provide front court depth. One of the main reasons he signed with the Hawks, however, was the opportunity for more minutes. It certainly wasn't for the money since he is making the veteran's minimum. The Hawks only had three serviceable big men in the rotation and desperately needed the help. So how have the Hawks rewarded this choice? By playing him at least 10 minutes less per game than any other time in his career, excepting an 11-game stint with the Nuggets. I'm sure he's not exactly thrilled considering he averaged 19.6 minutes last year on a Cavs team that was better than this Hawks team.
Out of the 6 teams with a better record than the Hawks, here is an example of players who are playing more than Joe Smith: Brian Scalabrine, Shelden Williams, Kyle Korver, Ronnie Price, Joey Graham, Anthony Carter, Anthony Johnson, Josh Powell, and Jawad Williams... to name a few.
This is troubling because given the example this sets, why would any other older veteran want to sign with the Hawks after being waived or even in the offseason unless we offer them more money?