Just a note, having a website made completely in Flash isn't going to be helpful for search engines. It's also going to annoy 2% of internet users..

.swf's (compiled Flash files) are generally small, as art is done with vector graphics and not bitmaps (unless you put in music, although I suggest that you embed music separately if you wish for such). Animated .gif's are generally quite large from my experience, I'm not sure how compression works but a heap of bitmaps is definitely going to be less efficient than vector graphics.
As an example, a 15x15 grid of hexagons (just an example, I don't expect this to be your background) is over ten times larger as a .gif/.png than as a .swf (drawn by maths - just a file I had lying around, and it can draw up to 50x50..). Also, the compression for the .gif makes it look somewhat ugly =.=
This balance obviously changes with smaller and more complex images though..