Tom Thibodeau has been trying to get a Head Coaching job since his final years in New York. The Don Chaney job, the one where he got passed over for. That's what 8 years ago? He's been trying for a job ever since then, and he's failed ever since.
Thibs has improved his reputation as well as humanly possible here in Boston. There's nothing left for him to do, or to prove, here. In contrast, by moving to a new club, he can reinforce all the positive things people think may be true about him and improve his reputation exponentially.
Moreover, he needs to make some friends. The man has no friends. He has no Ed Stefanksi (Jordan), no Presti (Carlesimo), no Ferry (Brown), no Bird (Carlisle). He needs to build connections with current + future front office personnel because he quite simply has none at this moment in time. Nobody is willing to go out on a limb for him and that's unlikely to change any time soon.
All that time he spent in NY and Houston, and there's nobody there who'll go to bat for him. Plus here in Boston, it looks unlikely that there will be any new GMs being generated in the near future, so no connections here either (outside of Danny). Moving to a new team allows him to build those connections. Staying in Boston does not.
Thibodeau has already done as much as he can in Boston. Staying hurts his chances at advancement.