From a longterm perspective, most fans would probably make this trade, considering the age of the respective players (Rodgers is 30, Brady is 37).
For this season, as Faulk poses the question, the Patriots would probably still stay in title contention. It would be hard to argue that the Packers would be better off than the Patriots, as Patriots coach Bill Belichick would find a way to acclimate Rodgers into his scheme quickly.
From a skillset perspective, the Patriots would be getting a more mobile passer in Rodgers. Brady has improved his mobility in 2014, but Rodgers is on a whole different level in that regard. In addition, while he attempts passes 20+ yards down the field at a similar rate to Tom Brady, he completes those throws significantly more often. You could argue that Rodgers' supporting cast has something to do with the huge discrepancy in those numbers.
Rodgers' intermediate game isn't all to similar to Brady's, as Brady's velocity and accuracy is incredible on those throws. If the teams made this swap, you would probably see the Patriots use Rodgers similar to how the team used Jimmy Garoppolo in the preseason. That means lots of short, quick throws and reads, utilizing roll-outs with shots down the field mixed in.
While I think there is a worthy argument for Aaron Rodgers being the best quarterback in the league, I wouldn't make the swap for this team. That being said, it should be a fun pre-game debate: how would you see a Brady and Rodgers swap working out?