Being gay is just a small part of who you are. You're not in the closet necessarily, but you don't wear your sexuality on your sleeve. It would be like taking pride in being a brunette or wearing blue jeans over khakis. Gay guys just try to live their lives as they always did. If asked, they'll talk about relationships or being gay, but they usually don't see it as something important enough to constantly bring up. Gay guys don't try to adopt exagerrated mannerisms or personas like gays are shown on tv, they're happy with still being themselves instead of signaling to everyone near and far.
A faggot is someone who has turned their sexuality into an agenda or political tool. They're the kind of people who after coming out will adopt a "gay voice" and flouncy, effeminate mannerisms because they WANT every stranger within earshot to know who they fuck. They call regular gay guys who don't act like whores and attention seekers "ashamed of themselves" a faggot will let you know they're gay within five minutes of meeting them, and be on the lookout for ANY slipup, real or imaginary that they can use to paint you as an evil homophobe. A faggot is usually an extreme leftist, right leaning, libertarian or moderate gay guys don't exist to them except as "traitors to the community" a faggot is all about "supporting the community" something the normal gay guys have become thoroughly disgusted with and abandoned long ago.