The amount of memory on a graphics card is generally not going to give you additional performance, unless the onboard memory of one card is insufficient for the task at hand. What more memory on your video card will do is allow programs to store larger amounts of data onboard. This is useful for high resolution texture packs in video games, video games that have a huge amount of textures or other productivity oriented tasks such as GPU rendering in a 3d modeling program, analysis programs that may be designed to run large data sets on GPU resources etc.
You would probably want to replace your video card quite a bit before the majority of PC video games require more than 2gb of memory at typical high end resolutions and textures.
The 270x is the same thing as the 7870, the GTX 760 is more on par with the AMD 7950. The 760 should give you better performance even with 2GB of VRAM. You will want more than 2GB if your running a resolution higher than 1080p or plan on running something like Skyrim with a lot of visual mods.
Overall the gtx 760 is more powerful than the r9 270x
but if you are gaming on a higher resolution or more monitors go for the r9 270x since it has more ram making it better for multi monitor support and higher resolution. However if you are only going to play on one monitor at 1080p you should definitely go for the gtx 760 since it annihilate the r9 270x. The gtx 760 also comes with cool features like physx and shadowplay
Which are very nice touches, I would suggest going for the gtx 760 personally.
keep in mind the R9 270x overclocks like a mofo... Without any heavy tweeking I have my memory at 6500mhz and my clock speed at 1200 mhz running at a cool 37 degrees...
My vote is the r9 270x as I can speak from experience... I am running the XFX version and the stock cooling is amazing so I could prob push it even further although I have absolutely no need as it is maxing out every game I own with ease.