You should use the Adobe Media Encoder and make sure that you have selected a high bit rate, best quality and two pass encoding. I don't know what Render cue Animated -h.264 means because there is no such codec. In some versions of AE you can choose h.264 Quicktime but you should never use that format. Apple has abandoned development of H.264 Quicktime and it never was very good. Only the AME gives you the option to do multi pass compression and that can greatly increase the quality of the render without effecting the data rate. You could also increase the data rate to either "high bit rate" or 20 or even 30 if the video is playing from a computer. If it is streaming from a server keep it to 20 or 25. You can always test the system used for playback to determine the best settings. I'm also seeing a color difference. You solve that problem by using color management. The last suggestion that I have is that you use 29.97 fps (NTSC standard) if you are in the US. You will have a lot less problems with motion artifacts at that frame rate. Solid colors against each other are always going to have some compression issues but yours look pretty good. You'll never get the same edge detail with moving things like text if you compare lossless codecs with I Frame (MPEG) compression but you should be able to get fairly close.
It is very important to realize that you cannot judge video playback with a still frame. The video has to be playing back on the intended device at the delivery resolution and frame rate. I carefully looked at the sample you gave and the only difference between the two hl.264 samples is the color saturation. The edge aliasing is pretty near identical as you can see by this composite. The best suggestions I can give you are to make sure you understand color management and turn it on, enable 2 pass compression in the AME and jack up the data rate. Design wise I would throw a little texture in your background. It can be very subtle and you'll get better edge behavior when you start moving type around on the screen.
Hi and thanks
Good input. Still a bit blurry, but a little bit better anyway.
This customer has an old system for displays so I´m very limited to relativity low bitrates.