Both have their good qualities. The Honda is smoother, a bit more powerful, and better on gas. They also seem to be a bit harder to get parts for, a bit finicky with the carbs, and more complicated.
A Briggs will have less power, your neighbors from 6 houses away will hear it, and they aren't the best on gas. They are also very simple engines, parts are everywhere, and they can run for decades.
Most of my OPE is Briggs powered, and many of them are over 20 years old.