all games are not created equal

They are pretty much the same game.  One big difference though; Quake Live is totally free and can be played from your browser.

Quake Live is meant for everyone and not only the hardcore FPSers.  This includes even those who may be totally new to FPS games.  Before you start playing, you have to complete a training course and a one on one match with a bot to determine your skill level.  Then after that, whenever you play online, Quake Live will try and match you up with people of your skill level.  Aren't able to rocket jump?  Have trouble keeping the lighting gun on your target? No problem!  The players you'll be put up against may have the same problems =)

Yea, the tech running Quake 3 is old.  The graphics are clearly dated.  But who cares, it's still a lot of fun to play!  And now, it's right at your fingertips, no matter where you are.

Now that I got to spend significant time with each game...

Quake Live currently offers a better game service, I feel.  I'm not gonna be picky about wait times and beta access and such cause that's unfair, but I like the mechanics and style of Quake much more.

Quake is faster paced than Battlefield.  While Battlefield offers vehicles, it doesn't offer much in variety of weaponry.  In fact, BH is a pain because you have to earn points within the game to buy weapons for your character.  Sometimes you even just RENT the weapons for a few real-world days.  This puts you at a serious disadvantage when you just start the game.
BH does allow you to customize your character with a variety of features and clothing, so thats a plus.  However they don't seem nearly as unique as Quake's characters.  However neither game really has a serious graphical advantage over the other.  They're each stylish in their own way.