If you're current monitor is connected via HDMI or DVI, then you should be getting HD resolution (it wont need to be downscaled to achieve the required resolution), though obviously not 1080p. I would expect there to be no difference, beyond an improvement in the quality of the monitor ofc.
As for getting full 1080p, it'd have to be upscaled a little more but the loss should again be minimal. Personally, I think a more important question is whether you'll notice the difference. Unless you plan on getting a 42" monitor or larger, I'd be surprised if you could tell the difference between a 1080p source and a 720p source from half a dozen feet, let alone a 720p source that's been upscaled.