To the OP .... As someone who moved to Atlanta in 1990, and lived thru the Braves 1991 - 2005 run, I will say the answer is the PATRIOTS.
Even native Atlantans admit that their "run" is extremely tainted and feels a little disappointing because of only winning just the one World Series. For the most part of their 1991-2005 run of excellence, the Braves had the best staff in baseball: Steve Avery, Glavine and Smoltz early on and then Glavine, Smoltz and Maddox later on. For a good part of their 1991-2005 run, the Braves were rolling out THREE Hall Of Fame pitchers (Smoltz, Glavine and Maddox). Think about that .... not 1, not 2 but freaking 3 HoF Pitchers..... and they won just 1 World Series. In addition to the 3 HoF pitchers, that Braves team will have 1st ballot Hall Of Famer Chipper Jones, and manager Bobby Cox.
When the topic of the Braves "glory years" gets talked about down here in Atlanta, the story will eventually turn to how such a loaded team could only win 1 World Series.
Contracts that to the Patriots who are actually putting Vince Lombardi Super Bowl trophies in their trophy case....... and it's easy. The Patriots.
A better comparison to the 1991-2005 Braves would be like the 1976-1985 Philadelphia 76ers ... a loaded team, played for many NBA Titles ... but could only win 1.