Having 2GB of VRAM will suit you just fine playing anything on ultra in the case of the GTX770 1080p. Getting let's say the 4GB windforce 3x Edition
http://www.newegg.com/Product/Product.aspx?Item=N82E16814125462 will be considerably better for a 3 monitor setup.
The CUDA cores help a lot. To understand this better, lemme explain something.
This example is a little on the oversimplified side, but it should give you enough to get started. After which, I'd suggest some of the fine articles on sites like Tom's to get more of the specifics.
If you think of your video card like a freeway, then CUDA cores would be analogous to the number of lanes in the road, clock speed would be the speed limit, and memory interface would be the number of lanes for exit/entry ramps. More lanes means more cars can be moving on the freeway at any given time, the higher the speed limit the faster any given car is moving, and then if there's say 2 exit lanes instead of 1, you can have more cars getting off the freeway, same as you can have more cars coming onto the freeway if there are say 2 entry lanes instead of 1.
Again, it's a bit of an oversimplified example, but it should be enough to get you started in understanding some of the articles explaining these things in greater detail.
In the end, the specs are irrelevant. What is important is game performance so you ignore the spec sheet and look at game benchmarks when choosing a card.
Now it's not to say that some games like Crysis 3 Might benefit of that extra 2GB of VRAM, (cuz it will), but if you're planning on gaming on an 1080p resolution only, no 3 monitor setup, 2GB will do just fine.
EDIT: Also, I should note that the GTX 770 model outperformes the 670, it being in the 700 series and having those amazing specs. Not only that, but it's a little bit cheaper, so I think IMHO that it would be the better choice.
http://gpuboss.com/gpus/GeForce-GTX-770-vs-GeForce-GTX-670
Then if you want to future upgrade, just SLI with another 2GB GTX 770 and you get 40% more performance than a GTX TITAN, which should say something considering the price to performance ratio.