You are probably seeing compression artifacts. The more you compress harder it is to eliminate these problems. Your animation codec quicktime is only lossless if the quality slider is set to best. H264 out of After Effects should only be used for quick proofing, not for delivery.
Any GOP (MPEG) compression scheme needs multi pass compression to do a good job. You'll have to use another app for that. Adobe Media encoder, Quicktime Pro, Sorenson Squeeze, HandBrake and a bunch of others do a much better job, but you have to know how to set them up.
You also may run into gamma problems. Most can be easily addressed by selecting the proper color profile settings.
I usally export the project totally uncompressed and then I do the final encoding-compression in Adobe Media Encoder but the problem remains.
I have every preferance to "Best" and "High".
Yes, I get artifacts...
About the Gamma problems, where can I select a color profile? In Adbe Media Encoder?
I noticed that when I select to manipulate the color settings from the player source (Windows media player, nero player) and not form Nvidia Quadro FX 4800, the problem solves. So the matter is that it looks bad on my monitor and in reality....