1. Dash cams are fixed and (usually) only see what is happening in front of the police car, which is normally on a public right-of-way and therefore where the public could also observe and record*. What happens elsewhere, like when an officer goes inside a private residence, isn't captured by dash cams. A body cam on the other hand would frequently be recording events that are not occurring where the public can see, and this is a significant difference for accountability.
This is the aspect that worries me. Privacy goes out the window with body cams. Anyone close to the "suspect" can get caught up in the same video, whether they have anything to do with it or not. As the parent points out, dash cams are used in public places; but body cams would be able to go into private places.
We know how well governmental bodies do with protecting private data (that is to say: poorly); imagine someone stealing a video about a controversial event, and there's your face in the video. You can get implicated by association, even though you just happened to be in the wrong place at the wrong time. Do you think your boss would be happy? Or your spouse?
This is even worse if you are a public person, where there would be even more of an incentive to steal the videos.