The one big problem with full screen mode in YouTube player is that you cannot do anything else on the computer while the video is playing - you have the exit the FullScreen before initiating any other task.
There's however an excellent alternative to Full Screen mode and that's about auto resizing the video player to fit the browser window. When you resize the browser window, the dimensions of YouTube Video player will change accordingly.
Here's how you fit the YouTube video in the browser window:
Say a YouTube Video URL is http://www.youtube.com/watch?v=vr3x_RRJdd4
To watch this clip in a resizable browser window without switching to Full Screen, just rewrite the above URL a http://www.youtube.com/v/vr3x_RRJdd4
In other words, just append the YouTube video ID to this URL www.youtube.com/v/VIDEO_ID and stretch your browser window. You can then easily switch to other tabs, open new applications or do anything else while the YouTube clip is still playing in the maximum possible screen area.