My converted videos look bad... what am I doing wrong?
Taking a closer look at the footage now that I'm at home, it's mostly the level of activity and motion in the shots.  Even in the Sony video, the quality absolutely falls apart any time that there is high motion and unstable video.

The way video compression works, it depends on predictable and smooth motion to achieve good compression results, especially for on the fly recording like a camera has to do since it can't look forward in time and see how things are going to move any further than it can buffer, which is generally not going to be very far (probably sub-second).

There are two places you could get quality loss.  The first is in the recording itself.  If you aren't recording at sufficient data rates, then if there is too much motion, fine detail is going to get lost.  Uncompressed video is absolutely massive and even most video cameras record with moderate to heavy compression already applied.  If your source footage lacks detail (and isn't out of focus) then you need to work on stabilizing the camera to ensure it can better compress the video.  There are a number of different ways you can shock mount a camera to help provide some stabilization, though they also generally add weight.

If the source footage is ok, then the problem lies in the final compression for web/end-user consumption.  The fundamental problem is still the same, but we now have the option of using digital video stabilization to help make movement less random.  We also can use VBR 2-pass compression since we can now look ahead and see how thing move in the future as well as the past, which allows for far more accurate compression and more efficient use of bandwidth.

Additionally, some of the footage could use color grading to enhance the contrast.  Ideally footage should make full range of luminosity (brightness).  Your blacks should be dark black, but without obscuring detail in the shadows.  Your whites should be bright, but without fine detail bleeding in to pure white.  You can start with a white point and black point, but really, using something like curves is the best way to tweak the video to make the most use of the available range of luminosity.  This can also be quickly, but sloppily accomplished by increasing the contrast, but results often look far more artificial that way.  Color grading is best learned by doing it and it can vary throughout a clip.  It takes time and practice to get good at it.