"A celebrity is one who is recognized where ever he/she goes."
I'd say that's a better definition of "famous" (person).    That would include the infamous, as well (Osama Bin Laden, Hitler, Charles Manson, e.g.).

I guess technically, a "celebrity" is someone whose existence or presence is "celebrated" (especially by the mass media).    "Celebrities" tend to be given "the red carpet treatment" and maybe treated as "VIPs" ("very important people").   
If they're given media attention and VIP treatment along with honors (a nobel prize, Oscar, Grammy, critical acclaim, Heissman (sp?) trophy), they're considered "stars" (especially in sports or entertainment). And if they are followed by millions of adoring "fan(atic)s," they're considered "rock stars."

I think today's TV (and associated infotainment) media generally use the term "celebrity" pretty exclusively for people in sports or entertainment whose names and faces are widely recognized (for the moment, anyway), but not so much for people much more highly respected and honored in fields like politics, business, science, religion, literature, etc.

It's not unusual to see/hear a statement in our US mass media such as: "The magazine issue features President Obama, The Pope, Bill Gates, Stephen Hawking, John Grisham and a number of celebrities, too."

Kind of like the statement I one heard on a TV "news" show: "The President is a real rock star!"