Best FPS game you have played

They're different, but largely equivalent in the points above.

As a story, Half-Life is better, since Q3 didn't have much of a story.
As a single-player experience, Half-Life is better, since Q3's singleplayer was just a training mode for newbies.
But as a pure shooter, Quake 3 is miles ahead of Half-Life, and that's what it's all about.

Look at where the HL series is now compared to Q3A? The original HL is almost completely irrelevant as anything more than a building block for better mods/games, and the sequel isn't much different. That's the thing with singleplayer games - you can choose to replay them every now and then if you're a huge fan, but there's never any challenge to it to keep things spiced up. Meanwhile, Q3A is still one of the most popular competitive games in the genre (along with other classics like Unreal Tournament and Counterstrike) and thanks to gamer-focused (as opposed to purely consumer-focused) initiatives like Quake Live is still a very popular multiplayer game.

And yes, there is indeed a reason this approach is considered outdated now. That reason is that the current gamer can't handle these kind of uncompromising games. There are no bells and whistles, no achievements to earn of gamerscore to raise, no classes to buy, no perks to unlock, and the graphics are extremely basic compared to modern shooters. Today's market is a market for console shooters, Call of Duty clones, plodding cover-based gameplay, achievements, cinematic storytelling, quickscopes and ironsights, maps and indicators, and graphics über alles. No big developer is interested in making the next Q3 or UT because the average modern gamer wouldn't be able to keep up with it, and all the big names do is cater to the average. Call of Duty is what the masses want so Call of Duty is what the masses get. There are attempts at capturing that old-school feeling every now and then (Painkiller was a good one - I didn't care much for the singleplayer but the multiplayer was solid as a rock), but none have really surpassed Q3A.

You can bitch and moan about how outdated games like Quake are, but that isn't any kind of argument. I get it, you like it when you're told a story and when you don't have to compete with real people. That's perfectly valid. I'm not trying to tell you you're wrong for enjoying Half-Life. I'm just telling you that you're wrong for saying Half-Life is a better shooter.