Water, mud, road grime, worms, etc. leave the tyre at a tangent (a straight line intersecting a circle). Notice how high up on this bike the mud has gone! It has even been deflected onto the back side of the fender! This is why a fender or mud flap is needed. The initial installation may protect the rider's face and torso, but his/her feet still get wet.
Also, it is very important to keep the water, mud and grime off of the drivetrain components. Grime will shorten the chain and cog life substantially. A rear fender flap is only needed if you plan on riding with friends so you don't spray water in the eyes of the rider behind you.