Mountain bike helmets usually feature a visor, which is one of the most obvious distinguishing features when comparing an off-road helmet to a road helmet. The visor adds a little extra protection from the sun, weather, tree branches and, in less than ideal scenarios, the ground.
Road helmets forego visors as it helps drop weight and improve aerodynamics and ventilation. If road riders do need a little extra protection from the elements they will opt for a cloth cap instead.
Whether you want a peak will depend somewhat on the speed and protection you seek, but in reality, it's more likely to be decided by the aesthetic of the helmet. There are no rules but we generally opt for a peakless helmet when wearing lycra and a visor if we are in baggier clothing.