You may remember the Saints stopping a game against the Chargers earlier this year when Brees broke the long standing record, passing Johnny Unitas for most consecutive games with a touchdown pass. I thought it was a little over the top. It's just a record. In the passing league that the NFL has become, it won't be long before the record Brees newly set will be broken. In fact, it could happen next season.
Tom Brady is only 11 games behind Brees' new record. Brady has thrown a touchdown pass in every game since September 12th, 2010. His streak of 43 straight games (third longest all time) is the only active streak in the NFL above 20 games. The next longest streak is held by Brett Favre with 36 straight games.
Brady could pass Unitas for second all time in Week 17 this year if he throws a touchdown pass in every remaining game. Interestingly enough, Peyton Manning's longest streak is only 27, which he set in 1999.
Tom Brady is more than capable of catching up to Brees and breaking it in Week 6 of next year. Except, I doubt that you will see the Patriots stop a game in order to acknowledge the record. I'm sure you won't even see Tom Brady care all that much about it.
The only thing that the Patriots care about is winning Super bowls. If Brady never throws another touchdown pass the rest of the year and the Patriots win the Lombardi, I don't think Brady would care at all.