Multi-VGA output support : HDMI/DisplayPort ports
- Supports HDMI with max. resolution 1920 x 1200 @ 60 Hz
- Supports DisplayPort with max. resolution 2560 x 1600 @ 60 Hz
Which means you cannot get 120 Hz at your 1920x1080 resolution with HDMI cable. You need to use display port output from your motherboard. Said that, I think you'll need a discrete card to achieve 120 Hz on your monitor.
$500 dollars can get you either HD 7970 Ghz Edition or GTX 680. It's pure preference whether you prefer nvidia or AMD, because even though GTX 680 is a bit faster without overclocking, when both are overclocked, HD 7970 catches up with it.
GTX 670 is about on par with regular HD 7970. It's around 7% slower than GTX 680. HD 7950 is below that. They both support the monitor, and Nvidia card doesn't need third party software to run 3D on it, while AMD needs it.