I suspect that the issues were relatively simple. Branch was available. Brady and Belichick wanted Branch back. Moss was then excess, since Belichick had no intention to extend his contract.
Branch and Moss are really incomparable.
I think Branch has fewer catches of passes thrown 20 yards or more in the last 2 years than Moss does this season alone.
Branch gives the Patriots some insurance and another option on the field for a dink and dunk offense that relies on YAC of some pretty crafty runners (like Welker) to keep defenses honest.
That, plus the fact that Hernandez and Tate can catch the occasional deep route pass and/or pick up big YACs might be enough to keep the field a bit more open than in the Reche Caldwell era.
Although I'm typically of a "you need a deep threat" mindset I'm actually optimistic that this new look offense is going to give some teams some major challenges and keep DBs playing deeper than they might normally against a team without a deep threat.
In an odd way this makes the Jets major investment in CBs to respond to a Randy Moss offense look like a waste of money. I think Revis and Cromartie do better against a guy like Moss than shifty guys like Welker, Branch, Tate and Hernandez.
This may be the biggest under-appreciated aspects of the trade. The Jets defense was unquestionably built to respond to a Randy Moss lead attack. That's no longer what the Patriots are and they may have some difficulty adapting.
Branch's post season production is nothing to sneeze at either. His overall numbers might not be eyepopping but we all know that when crunch time comes he can make the big catch.