mfratt, I started out talking about C-41 chemistry, an old Jobo Tetenal C41 Press Kit, specifically, then I switched to a discussion of RA4 printing. Now back to film processing, which gets this thread back on track (not that I mind it wandering, especially when I'm the one who instigates it) ...
The most often cited advantage of using the Kodak C41 chemicals that I have read on this forum is that the bleach and fix are separate steps, and this is supposed to yield more stable negatives with less chance of color changes (I think) due to under bleaching or under fixing. I think that's the gist of it, but there's probably more to it than that. There are many excellent posts by Photo Engineer (PE) and others who discuss this in far more technical detail than I ever could.
A disadvantage of using (or attempting to use) Kodak C41 chemistry is that it isn't available in small quantities, there are at least three different formulations intended for different types of processing, all components are separately orderable, and figuring out exactly which components you need to combine together is challenging (for many of us). There are myriad lists of the CAT numbers you need to get, all different (slight exageration).
Even more difficult is sourcing all the chemicals you need in small enough quantities to be reasonable. The bleach ships with a hazmat fee. Some merchants won't ship it at all (B&H). You may not be able to source all the components from one merchant. The merchants who sell it retail are often out of one component or another. But many people say that if you can sort out all the myriad CAT numbers and sourcing issues, Kodak is the best way to go for C41 processing. That's obviously a subjective evaluation.
I plan to switch to Kodak chemicals for my C41 processing after I use up the Tetenal kit I have now and the Arista C41 kit I recently bought.
Mine is stored at 70 F or cooler.