Fact: EVE has been around since 2003

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Now, I own and actively play both games. I have played TF2 since August 2009 and EVE since Feb 2010. I have more or less the same hours put on each game (Nearly 1000 hours). Both are awesome games, and comparing them is quite hard since they're not the same genre, and they're oriented to different publics.
TF2 Is a great, extremely fun multiplayer shooter. You'll never get tired of it with the new content and the lols going around. Definitely worth getting.
EVE is a space simulator-somewhat-mmorpg style that offers a superior game experience, with a lot more than just laughs. You can get to be pretty big in EVE, if you know the ways, and some of this ways are just like real life. It's a slower paced games, and requires lots and lots of patience. Believe me, sometimes doing an operation in EVE can take up a whole night, and you can hardly quit in the middle of it.
Like everyone else mentioned, EVE has a monthly fee, that can be covered with ISK (In game currency). You either get a 14 day or 21 day trial. Is it possible to get such amount of ISK in 14/21 days? Yes it is. It's really hard, but not impossible. Other than that, eGO can offer training plans to get to making that much ISK in a month or two.
Notice I put a lot mor emphasis on EVE, and that's just because it's a deeper game.
Whatever you pick, I'll see you there!