If your model has coloured triangles, you might find that the binary STL preserves the colours, while the ASCII STL loses the colours. Some tools have a way of putting colour information into the binary STL format, which isn't possible with the ASCII format.
This is what fred_dot_u experienced in his answer. In this case, it'll make a huge difference which one you use, but you'll only find out when one goes wrong. The same is true of the slicer, which may have a bug reading one of the two kinds of STL but not the other. One exporter may have a bug that the other exporter doesn't. I don't know about Meshmixer specifically, but some tools will have completely different code paths for exporting the two formats.
There are a couple of important exceptions here: The number of triangles and the dimensions of the printed model will stay the same.
That's to say, if you take the exact same model, save it as a binary STL and as an ASCII STL, the binary STL file will take up fewer bytes on disk. The two formats contain the same information about the model, but the binary format is much more compact, so it will produce smaller files from the same part but they should work the same.
