Firstly, IMO there is no "best developer." That said, the main difference between 510-Pyro and Pyrocat-HD is the stain color. The former leans more toward green/yellow, while the latter leans toward a brownish color. But, I've used both formulas over the years (as well as several other pyro-based developers) and have found that the stain color obtained is somehow related to the specific film developed. No idea why, but that's what I've observed in my own use.
Another difference is how they'll print, if you print in a wet darkroom. The 510-Pyro negative will tend to hold back the high values (not as much as PMK, in my experience) while the Pyrocat negative will print with a fairly straight line to its curve. For me, Pyrocat negatives seem to provide an increase in contrast with modern VC papers, but it's easily remedied.
Oh, and 510-Pyro will last just about forever. Opinions vary widely on Pyrocat, but I've had it die within one year of mixing. It does keep better if mixed in Glycol, but I've had that, also, within a shorter timeframe than what I typically see reported. And, what has always bothered me about its "death" is that it's a "sudden death" (like Xtol) and you don't know until your negatives don't develop properly! I still use Pyrocat because I think it's a great formula, but I tend to mix it in small quantities every six months and toss what I didn't use.