The problem with underwear is bunching and seams. If the fabric bunches, it causes pressure points that will press, chafe, and ache as you ride. If you perspire, they can burn. Same thing if the fabric has lumpy seams (common on y-fronts in my experience). Low-quality bike shorts can have the same problem if the chamois is a composite and has seams (example: old 1986 Performance Century Shorts) or has lumpy seams in the outer short. You want the bike short tight to provide some of the same type of support you expect from a jockstrap, but really some of that is provided by the front part of the bike saddle. But when you sit on a road bike saddle, you want nothing sensitive to be wandering about and getting caught.
Here we're mainly talking about the tight, Lycra shorts, not teh baggy, cargo-like shorts often suggested for mountain biking. They really do serve the purposes of conventional underwear, without introducing the issues that can make a road ride and the four days that follow it total hell.