This is only peripherally my area of expertise in a previous occupation; that said, let's use film as an example:
Film editing is taking already-shot footage and cutting some out, cutting some together, etc. This is clearly an oversimplification but just as an analogy.
With sound editing it's a similar process in that the sound has been recorded and the editor matches it to other media in all the right places.
Sound Mixing is the process of engineering the recorded sounds; e.g., the dynamics, compression, signal-to-noise ratio, etc.
Both jobs are difficult because they require technical savvy AND artistic talent. This is a very simple description of the differences between the two jobs. I expect an industry professional will post a more detailed answer at some point.
