Twice as many pixels, so more room to do stuff. Having two windows open at once is easier, and you see more in games. Like going from 720p to 1080p.
You'll either need a faster GPU, or to turn down settings or live with lower frame rates, because you need more graphics power.
It should just work, if your GPU has a displayport or dual-link DVI connection. HDMI is doable, but you usually need to set a custom resolution.