It really depends what you are trying to do with the monitor. If you are using it to run applications, the higher resolution will help a bit. You'll be able to see slightly more. Realistically though, the advantage is pretty minimal.
If you are using it to watch HD video, the 1920x1080 monitor will have a slight edge. To make 1920x1080 video fit onto a 2048x1152 screen, it will need to be resized. This will cause slight blurring (probably not so you notice) and more importantly, it will take a lot of CPU/GPU to do the resizing. That's processing power that can't be used for other things.
As to the second question, you want to run the LCD at its native resolution. It will always display 2048x1152 pixels so feeding it a smaller picture means that picture needs to be stretched out to fit. The scaler in typical monitors isn't all that great and it will likely become fuzzy.