I remember a few distinct phases from a highly subjective PoV in NYC/NJ--After beating the Rams, the Pats were gritty, scrappy underdogs, coached by a near-genius who took a middling team with a boy-band cute QB all the way. The book on Brady, as late as the 2003 playoffs, was that he was cool under fire but was never asked to do much beyond what BB and Charlie Weis carefully laid out for him--"He's a system QB!" It wasn't until the 2004 team was dominating that many people outside NE and/or who hadn't been watching very closely recognized that something special was happening--and that Brady was a huge part of it.
Following 39, I think they were seen as a highly respected dynasty. I also recall this as the period when the 'TB or Peyton' debate began in earnest--Brady had the rings, the perfect playoff record, and an emerging rep as the best big-game QB since Joe Montana while Manning had the passing stats and a rep as a brilliant regular season player who wilted in the playoffs, especially against New England. Some of this began to shift after the Pats lost to Denver in 2005 before really getting interesting after the flu-immersed loss to Manning's Colts in the 2006 AFCCG. That loss, however, also had the curious effect of galvanizing the pro-Brady camp, who began to argue that Brady accomplished much of what Manning had with vastly inferior talent at the skill positions, Corey Dillon and, to a lesser extent, Troy Brown notwithstanding.
Then came Spygate and the revenge-for-Spygate dominance of 2007 followed by the crushing loss in 42, Brady's lost season, two consecutive playoff humiliations in 2009 and 2010 and another heart-breaker against the NYG. I remember a lot of nonsense about how the Pats couldn't win without cheating in those years--as if the majority of 2007 wasn't played after Spygate. But from then until they beat Seattle in SB 49, I think most saw them as a consistently very good, occasionally great franchise while a few idiots muttered things about video cameras and taped walk-throughs. I think that's pretty much how it stands now--except the very good/great balance has shifted--the Pats are seen as a great team that every few seasons manages to be 'merely' very good. The pinheads who bleat, "Cheatriots" have been thoroughly marginalized, it seems to me.