popkorn1,
Hopefully, someone will correct me if I'm wrong here but here's my experience. My move was from using composite on an old TV to component on a newer TV. I think you indicated that your TV has a line doubler. So, you've already improved on the picture a great deal (no vertical shaking anymore). Like the others have said, your moving from s-video to component cables would improve your color quality (more vibrant colors and less bleeding, I think).
In my own experience, upgrading to an HDTV and using component cables made a huge difference (very very noticeable). Extremely vibrant colors, clear and no vertical shakes from interlacing. You do have to tell the gamecube to display in progressive scan mode - and it's possible to have the gamecube transmitting the normal (non progressive, by not pressing B at boot up) signal through the component cables. Sometimes I run the games in progressive; other times not. To be honest, I can't tell one bit of difference in the two (both ways using component cables, mind you). Don't know if this is due to the line doubler or not or simply that the gamecube really isn't taking advantage of the higher vertical resolutions of HDTV. For instance, Metroid Prime looks exactly the same either way. I thought that at least the game would appear smoother due to the effective frame rate doubling, but I can't tell a difference there either.
But in your case, you already have s-video and your TV has a line doubler. So, I'm guessing that the only difference you will get with upgrading to component is improved colors (and possibly a little clearer ?).