I'd say (and I'm definitely not an expert, just an old timey lurker) that there was a culture on 4chan where the vast majority of posters considered themselves to be 'anons', specifically in /b/ where you're unable to give yourself a name. Other imageboards on 4chan do allow names, and even things known as 'tripcodes' to help protect the authenticity of a person's identity without a complicated login/pass system, but generally people who adopt names and trips are looked down upon unless they're proven authority figures or moderators.
When channers took to the streets, not just 4chan posters inhabitants but also members of other chan boards, the general sensation was that anonymity was part of the draw behind those image boards and was preferable in public demonstrations.
Basically: "4channers" doesn't have a real ring to it, and showing up in public wearing a mask that has nothing to do with 4chan only confuses the matter. Anonymous, however, is a word that most people know and understand the meaning of. By default, if someone is wearing a full face mask they are essentially anonymous until their true identity is revealed.
It should be noted that while the popular public image of Anonymous includes the Guy Fawkes mask (which was fashionable at the time of the scientology movements), Anonymous is more often characterized as a suit clad figure with hands out of sight. The face doesn't matter, and indeed many representations of the suit clad figure are entirely headless or simply a green, faceless head (green may relate to the four clover leaf symbol which is the 4chan logo, but I'm not a symbolism expert). I will however, gladly conjecture that the suit clad figure represents one of the slogans of the 4chan community which is "The Internet is serious business".
I think the general public awareness of "Anonymous" took off due in part to many factors. In 2006, Google acquired YouTube, making the existing site better known to the general internet community and serving as a launch pad for meme distribution in ways previously unthinkable. A college friend may have sent you to YTMND, but your grandmother linked you to the funny cat video on youtube. With memes gone mainstream, more people sought the root of this funny material and invariably the traffic of YTMND, SomethingAwful and 4chan swelled.
With more members, there were more lulz to be had and more people to carry them out. More incidents became publicly known (Pools closed, Internet hate machine, etc) through conventional televised media, drawing more traffic to the site. With the 4chan protests, even more television, newspapers and blogs reported on the existence of the so-called group known as "Anonymous", now famous for protesting scientology while wearing guy fawkes masks. Tragically, there was a bit of an outpouring of well intentioned protesters who had no idea what 4chan was or why people were wearing masks, and some people tried to wrangle Anonymous into being some sort of faceless champion for public causes.
I'm certain that public awareness of the ambiguous entity known as Anonymous still works for and against 4chan traffic, but the majority of 4channers seem to loathe the post-scientology "public champion" image Anonymous was stuck with. I haven't seen a whole lot of open support for any Anonymous activity on 4chan, be it online or in person, for a while. As others have posted, the original anons involved with /i/ very likely had little to nothing to do with the current state of Anonymous and the many little groups operating under that banner.
It seems that it was simply part of the 4chan culture that slipped out, was made a public label for 4chan's special form of internet vandalism, and then by the same oldschool media that had vilified them as the 'internet hate machine', accidentally christened them an internet anti-hero.
