Well as the difference is minimal then get the 2GB version. It wont be any faster but will possibly be better suited to future games.
Could you post links to the 2 cards so we can look over the specs in case you are missing anything ?
^+1. Current games practically never use more than 1gb on 1920x1080, but you never know what the future will bring. I don't think it's worth spending any more money on the 2gb version, but if you don't have to drop more cash, go for it.
The 1GB Ti card is your better option here, basically they are putting the extra memory on the standard 560 to make it look a better card than it is, just marketing.
Just to be clear here regardless of your resolution the Ti will be the better/faster card.
The card is not powerful enough to make games playable at higher resolutions where the higher memory amount might have made a difference.