If you set the 360 up to output anything other than 720p, it's going to have to either stretch the image out or shrink it down.
With your current monitor, it would have to shrink it down. Shrinking an image means you're going to lose pixel information, as there are physically less pixels in a 1024x768 (786,432 total) screen than in a 1280x720 (931,840 total) screen. I also can't comment on how all games would display the 1024x768 image. Some 260 games seem to stay in their 16:9 resolution, while others seem to change their resolution so that the entire screen is filled. What this means is that some games on your current monitor will be letterboxed (black bars), while others might fill the screen entirely.
With the monitor you're looking at getting, the native output resolution from the 360 (1280x720) will have to be stretched out if you want the image to fill your screen. This can either be done via the 360 menus (though they only allow up up to 1360x768 currently. This should change in the fall!), by your monitor's internal scaler, or by some combination of both. The other option on that monitor might be to play the game with a 720p image in the center of the screen with black bars around it entirely.
I use a CRT, so I just have it displaying a 720p image without problems. I'll try to post an idea of what output might look like at 720p, shrunk down, and also stretched up. I'll have that up in a few minutes