Who will end with more career passing TDs - Brees or Brady?

Brees.

Disregarding age, I think Brady does because the Saints are transitioning to a more run oriented team

Brees will be #1 in passing yards, completions, and completion percentage. He will also be #1 or #1 in passing TDs

Looks like Brady will.

Payton is transitioning to a balanced offense, with more emphasis on the run. Brees is more of a caretaker and coordinator on the field, and will be responsible for the TEAM getting the score. If that means Ingram or Kamara run it in, fine. Also, Brady has Gronkowski. So that means 1st and Goal from the 5 (for example), Saints likely run a couple times, the Pats throw to Gronk.

However, I think too much is being made of Brady's 500 yd performance last night. Yes, it was great.
But Brees showed vs Minnesota that if he has to, he also can still go nuts. He didn't have 500 yds, but he sliced the Vikings offense up like butter in the 2nd half, just as effectively as Brady last night against the Eagles.

TL DR answer- I think Brady will, because the Saints have other options and the Pats don't.

Probably Brady because they're going to throw it more the next couple of years.

Can't believe I'm saying that, but it's true.

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I think Brady will.

Brady is older, but he is still asked to do a ton and has looked good doing it.

Brees and the Saints may be moving towards a more rush friendly offense.

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Yah I don't project the Patriots deviating much from their RBBC/mixed pass selection approach that they've basically perfected for the past 8 seasons across 2 OCs (O'Brien and McDaniels), too effective and consistent and I notice the Saints offense is in a state of transition to a more balanced approach which was reflected by the statlines this season relative to the run and pass components-2017 was the first season that Drew didn't reach 30 TD passes since 2007. Will be an interesting QB career legacy situation for sure.

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Pretty much comes down to who retires last, and there's no way to tell

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This is the correct answer. The Saints may be going more towards a balanced offense, but if Brees plays a year longer than Brady, he will throw for more TDs. The Saints running more than the Patriots isn't going to sway their TD totals enough to compensate for an entire extra season. For instance, let's say Brady plays one more season and Brees plays two more. Brady threw 32 TDs this year. So let's just say he throws 30 next year and retires after the season. Brees would have to throw fewer than 30 TDs over two seasons not to pass Brady. And for reference, Brees threw 23 TDs this season in that balanced attack. If both QBs had 5+ years left in the tank, then the Saints going more towards the run would have a bigger effect. But both are in the twilight of their careers, so the only way that will matter IMO is if Brady and Brees retire the same season or Brady retires after Brees.

Brees is younger and plays inside.

Seems like Brees will he will probably play 2 more years than Brady.

I expect the saints will be targeting a top flight tight end and wr to help in the red zone.