Is there any visible difference in picture quality of 1080i and 1080p 42 inch LCDs if your cable signal is 1080i?

Probably not. All that means is that the I (interlaced) and P (progressive) scans are interpolated by the TV set and then displayed. All of the picture info should still be available for display.

In general terms, all the interlaced or progressive scan means, is how the TV handles the display on the screen where the progressive scans all the lines of video information progressively (one at a time, sort of a stacking one on top of the other like a computer monitor) to build a picture, while the interlaced scans the info by interlacing several between each other as the display builds.

You should be able to find more accurate info on the web since it'd take quite a bit of time to explain it here in easy to understand terms.