For me, Core Only means that the players are using only the PHB for character creation. I usually ask if people want PrCs in or out, and I've played with PrCs "out", for simplicity.
Now-days I would say Core Only also includes, say, DMG PrCs and applicable feats from the MM, but not Level Adjusted Races and the like. No sub-races, either (of which the elves are broken and the others usually just dumb).
I did this because I ran alot of college games where we drew players from a large pool of Folks I Don't Know. After the first game (seriously) I picked up two really good players that generally stayed with my various games through the rest of the time I was there, but you never knew who'd ask to join.
Why'd I do that because I didn't know the people? I knew the pool. Other folks were running games and I'd hear horror stories about the guy that wanted to play the Vow Of Poverty Feral Anthro-Bahleen Whale Monk or vampires or the guy that carried his Oriental Adventures book everywhere and wanted to play characters from it. Generally players who figured they could outsmart the game and the GM and "win" or whatnot ... usually they were outsmarting themselves more than anything, but explaining the rules to some guy intent on playing a mish-mash rule-breaking character isn't on my list of Things I Want To Do This Session. By stripping things down, people actually played the friggin' game ... some of those guys with the bizzare characters had never played a single-class core-race character, and found the game much more enjoyable for not being buttheads. Othertimes it just weeded out the buttheads.
And I think the game can be perfectly fun, interesting, and fulfilling without additional splatbooks. If I didn't, I'd probably not have bought it.
If playing with, say, my current group ... I'd be generally cool with them playing whatever they wanted to play, since I'd trust them not to be trying to be buttheads but have a keen concept they want to play.
I've had people who balked. Usually they came around to my way of thinking soon enough, or decided not to play. The guy who decides he'd rather not play a game because he can't have some obscure race/class/feat is probably the guy I was going to have "issues" with at the table anyway. It never made the game less-than-fun to play core.
--fje