As the marketing/PR guy for 13th Age, I thought I might chime in here with what it looks like from my side.
It was indeed a coincidence that 13th Age was announced around the time that WoTC announced Next. At first we were concerned that the 13th Age announcement would be completely drowned out, because Next was huge news, and would hold the attention of the fantasy RPG community for the foreseeable future.
It's impossible to say what might have happened if Next hadn't been announced when Pelgrane started the 13th Age playtest. What did happen Next-wise, though, was that people wrote blog posts and started forum threads to talk about what else might be out there in case Next didn't deliver the experience they wanted. As a result, for several weeks our game was mentioned in the same breath as Dungeon World.
We've gotten great buzz so far. We've seen positive 13th Age writeups and designer interviews on the top gaming blogs, and threads on the major forums. The game's been talked up on Haste, Geek in Review and That's How We Roll, and was featured twice on the cover of Kobold Quarterly magazine. We ran well-received demos at Gen Con, PAX and Metatopia, and got a shout out on Penny Arcade.
But we're still not anywhere close to stealing Next's thunder.
The game is unlikely to do so for a number of reasons. D&D is the hobby's strongest brand, and you'd go through a substantial list of other strong brands before you eventually got to 13th Age. We don't have anywhere near the marketing resources WoTC has (I work on it part-time on top of a day job and a regular gig at Kobold Press), and 13th Age is a quirky, designer-driven game that's not going to appeal to everyone. (Here are the grappling rules in their entirety: "We don’t want general grappling rules. If you decide to allow normal PCs and monsters to grab hold of people, try something like this: their puny grabs are like real grabs but they don’t prevent opportunity attacks, don’t provide an attack bonus, and only apply a –2 disengage check penalty.")
In the end, we really, really want Next to succeed, because when D&D does well, the hobby does well, and our game has a better chance of doing well.
By the way, if you want to hear 13th Age being run by one of its designers, all this week BJ Shea's Geek Nation podcast will be posting daily actual play recordings with Rob Heinsoo GMing for the group.