You can't think in terms of "all Hondas" or "all Briggs" being good or bad.
The good Hondas, the red and white ones, are by far the best engines out there. The cheap Hondas, the black ones, are garbage.
Briggs also has different engine families of varying quality.
Which one you ought to have on your splitter depends on your needs. If a splitter is going to be a commercial or rental unit, then the good Honda is the only way to go. But for personal use, I'd go with the Briggs. The inteks start just about as easy as the Honda, and easily better than the cheap Honda. I have the old style Briggs, on my cheap MTD 20 ton. It's not even an IC/sleaved engine, and it's 8 years old. No it won't last as long as the better Honda, but I don't need it to. If you are just splitting your own wood, you simply aren't going to put that many hours on the thing. I can do an awful lot of splitting on a single tank of gas.
I don't like the Tecumseh engine on spiltters. They don't seem to start as easy as their snow engines, and the carb/air filter with that 3" choke lever is too susceptible to being damaged by a chunk of wood falling on it. BTW, Tecumseh is not owned by Briggs. Their engine division was recently bought by some investment group and will likely be purchased by some other Equipment OEM.