NVIDIA vs. AMD
Nvidia Pros:
- G sync
- Shadowplay
- Gamestream
- Streaming to Nvidia Shield
- PhysX
- 3D Vision
- Significantly better than AMD in applications that can use CUDA acceleration
- Lower power draw, heat, and noise levels (for the current generation GPUs, this could change in the future)
- Holiday Bundle which includes 3 free games (this won't be forever, obviously)
- Lower failure rate (I'm not being a fanboy, I'm making an observation based on a lot of the reviews I've seen on the Nvidia and AMD cards)
- Green
AMD Pros:
- Mantle
- More VRAM (better for higher resolution gaming)
- Wider Memory Bandwidth (also better for higher resolution gaming)
- True Audio
- Significantly better than Nvidia in applications that use OpenCL (Bitcoin and Litecoin mining for instance)
- Bundled with Battlefield 4 (this won't be forever, obviously)
- Red
Now on to your questions about G-sync and Mantle. G-sync is essentially a piece of hardware built into the monitor that syncs the monitor's refresh rate to the GPU's refresh rate, instead of the GPU's to the monitor's. This essentially fixes the screen tearing issues that V-sync fixes, but it does so without causing input lag or stuttering. It essentially makes the game super smooth, regardless of the frames per second you're getting.
Mantle on the other hand is low level hardware optimization for AMD graphics cards. It should allow for much better performance from AMD cards in games the use Mantle, because the game code can interact directly with the GPU instead of going through a slow bottleneck like DirectX. Unfortunately, no one has released any real world performance benchmarks for Mantle yet, so we don't know if it's going to be the massive game changer that AMD fans are hoping it's going to be, or if it'll just be another marketing gimmick. I do have high hopes for it, and even if it doesn't bring significant performance increases, as long as it improves the performance a little, that's good.
Now as to which GPU to choose, that really depends on your price range and the games you play. Some games run better on Nvidia cards, some run better on AMD cards. The best way to decide to simply check out benchmarks for the cards in your price range, and make your decision based on that. They're pretty much neck and neck at all price ranges, although the high end AMD cards are pretty much sold out right now because of Litecoin mining, so you may be forced to go with Nvidia at the moment simply do to the lack of availability of AMD cards.
Hope this helps! Game on!