Yes, that’s the pixel grid of the monitor. Yes, a slight blur in AE will help out.
But you’ll get better results by downloading the video from YouTube (not shooting it off the screen) and then converting it for use in FCP. I don’t have the names of the programs that will do the downloading but I’ve done it before and a Google search will get you what you need.
Alternatively, if you want the best quality you can do what I’ve done in the past: Contact the YouTube poster and offer a nominal amount for a direct dub of the footage.