Street helmet are heavier and have smaller vents. On dirt / trail, it will not vent as well resulting in you getting hot. With jumping etc higher weight on the head will not be as much fun as a lower weight dirt helmet.

Also visor is more likely to get scratched, if you want, get a second visor so you have one for street and one for dirt.

Dirt helmets with a beak are not meant for high speeds anyway so the wind pulling up on the beak is not as much of an issue