When go to export a model using Fusion 360 or Meshmixer, I see that there are two options. Could the final model be affected by the format chosen at the time of saving?
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. 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 number of triangles and the dimensions of the printed model will stay the same.
There are a couple of important exceptions here:
I don't know about Meshmixer specifically, but some tools will have completely different code paths for exporting the two formats. One exporter may have a bug that the other exporter doesn't. The same is true of the slicer, which may have a bug reading one of the two kinds of STL but not the other. In this case, it'll make a huge difference which one you use, but you'll only find out when one goes wrong. This is what fred_dot_u experienced in his answer.
Some tools have a way of putting colour information into the binary STL format, which isn't possible with the ASCII format. If your model has coloured triangles, you might find that the binary STL preserves the colours, while the ASCII STL loses the colours. Whether this matters to you depends on what printing technology you'll be using. Most slicers can't use these colours anyway - and subsequently, ignore color information on import.
The ASCII STL format is older than the binary format, so you may find some very old software can only understand the ASCII STL files, but unless you're working with such old software, it's usually better to use the binary format. Smaller files don't just save disk space: they're also faster to process and transfer via e-mail or on servers.
You should always pick the binary option. ASCII files are larger and slower to save and load. There's no reason to ever use ASCII unless you are using software that is incompatible with binary files.
Could the final model be affected by the format chosen at the time of saving?
In practice, the model will not be affected by either choice. There are some subtle differences between the two formats, such as binary being able to store an attribute per triangle (which is sometimes used to represent colour), ASCII being able to store a "name" for the solid in a file while binary can store an 80-byte header containing metadata, binary being limited to 32 bits of precision while ASCII theoretically has the option to use arbitrary precision. However, for 99.9% of all use cases there is no difference, so it is preferable to use binary for its smaller file size.
I have experienced problems on occasion when using a binary exported Meshmixer model. The slicers used have been Simplify3D and Prusa Slicer 2.0 and possibly an earlier version. I've not attempted to resolve the problem other than to change that specific model to export to ASCII which then solves the problem. ASCII files will be larger but that's not a significant factor, in my opinion.
If you are using a program which fails to properly process a binary export, it's simple enough to overwrite the model in ASCII form.