I have been a huge PES fan for many years. Although, last year I was quite disappointed with PES, so I made a change and purchased FIFA 12.
Once the demos for PES and FIFA 13 was released, I gave them both a good trial, and I came to a conclusion...
If you already had FIFA 12 and you want more of the same action, go with FIFA 13 cos it hasn't improved much.
If you fancy a completely revamped footy game, go with PES, as it has improved in just about all areas.
I've noticed the problem with FIFA is, the teams all feel the same - it doesn't matter which team u are or who you're against, the AI always feels the same. The teams on PES all play different styles, bringing a bit more variety, this becomes much more of a challenge as it becomes more difficult.
Basically if you want a nice 'easy to pick up and play' game, buy FIFA 13. If you want a challenge and get your money's worth, go with PES 13.
Clearly I have reverted back to PES this year, I found FIFA too repetitive.
EDIT: I now have purchased the full version of PES 2013 and I'm still really impressed. Obviously, the biggest downfall with this game is the lack of licences. But if its gameplay that is mostly important to you this shouldn't be a problem. Although, you can purchase cheap unofficial updates for this game to correct all the names/kits/etc.
This game has become very realistic. But it's so realistic that you may have to be so patient at times, like playing a possession game and only just win a 1-0 victory. The AI has become a lot smarter, especially now that there are 6 difficulty modes.
I'm especially enjoying the master league. Despite the fact that people are disappointed with the whole Boots and Abilities add ons, I think it's an extra bonus to the game. You earn points by winning games, then use the points to buy abilities etc in the Pes shop.
I'm definitely impressed with PES 2013. I've not felt this way about it since PES 5. It has been a let down almost every year since the Ps3 was released, but now they seem to be back on track... And fifa may just be losing this battle for once...