Tomb Raider vs Tomb Raider I

Tomb Raider I is the original Tomb Raider game, while Tomb Raider is the most recent series re-imagining. I'd go with the new over the old personally, the original Tomb Raider has not aged well at all.

Tomb Raider is the prequel of the entire series. Tomb Raider I comes later in the timeline, and it's about you retrieving 3 pieces of an atlantean artifact called "Scion".
The story of Tomb Raider is mainly hinted in the old games in Lara's biography, in the manuals mostly.

Tomb Raider 2013 is amazing (LOVED IT!) and tells the story of how Lara became the kick-ass woman that she is. It's a kind of a prequel, or so I hear -- I haven't played the original games yet!

Tomb Raider I is the first game which I owned on PS1 and was awesome back then but hasn't aged well. Tomb Raider is a reboot of the franchise - and as much as I'm an older gamer who likes bigging up the classics it is basically a lot better in every way. Get both games if you can afford it or just the newer one if you can't.

The question you should be asking is how Tomb Raider I compares to Tomb Raider: Anniversary, and whether it is worthwhile to own both.

Tomb Raider (2013) and Tomb Raider I are unrelated, vastly different games. Tomb Raider (2013) is a prequel to the entire series and is a more modern game, so I would say start there and go back if you become a fan of the franchise.

You'd be better comparing Tomb Raider: Anniversary to Tomb Raider 1. I very much love TR:A, and if you're thinking of playing some of the old games, I'd suggest you to just pick up TR:A, because it's a full remake of the original TR 1.

this small documentary will help you.

All your history are belong to us: Tomb Raider