If you're referring to 1993, let's look at who his fellow pitchers would have been on each team:
Yankees: Jimmy Key, Melido Perez, Bob Wickman, Jim Abbott, Scott Kamienicki
Braves: Glavine, Smoltz, Avery (Don't even need first names) & Pete Smith
He decides to go to the better of the 2 staffs (the only Yankee starter on the team who could have fit in on that Braves staff was the amazingly underrated Key) with the intention of finally winning consistently (1 trip to the postseason during his time with the Cubs). Not only does he sign with arguably the best staff in MLB, he becomes their Ace, and leads them to postseason after postseason. Sure those Braves teams only have 1 ring to show for it, but it's beats having zilch. And of course eventually the Yankees late 90s run became >>> than the Braves early to mid 90s runs, but back in 1993, no one was predicting a Yankee dynasty.
One more thing, in 1993, the AL East wasn't the murderers row of ball clubs it gets touted as today. Of course there were good teams in it, especially the Blue Jays, but there weren't multiple 90 win teams consistently in the AL East then like we've seen the past few years, so I don't know just how much "exposing" the division would have done to one of history's greatest pitchers.