When properly formulated, I doubt if there'd be much difference between stop baths. They're just mild acids, after all. I have seen differences between brands in the intensity of the dyes used in indicator stop baths, but I don't know how Kodak and Ilford specifically compare on this score. Unless the stop bath is exhausted or very defective, I wouldn't expect it would have much of an impact on image quality.
Fixers also won't have much of an effect on image quality, assuming they're properly made and used. They do have other characteristics that might be of interest, though, such as speed of action, speed of post-fixing washing, odor, hardening or non-hardening, and delivery as a liquid or powder. Both Kodak and Ilford make several fixers that vary on several of these characteristics, so it's impossible to compare the brands per se. There are, however, endless threads here on APUG about the relative merits of rapid vs. non-rapid fixers, hardening vs. non-hardening fixers, and so on.
Note that some fixer characteristics, particularly combined with how the fixer is used, can affect image stability. Underfixing or improper washing can result in the image degrading over time. I've never heard of specific products being better or worse in this respect, but you can easily mis-use any product. This could cause problems if you switch products but don't adjust your processing accordingly. For instance, non-rapid fixers need more fixing time than do rapid fixers (as you might guess by their names), so if you switch from a rapid fixer to a non-rapid fixer, you must extend your fix times accordingly.