One problem with a nose wheel as Silvaire1 mentioned is prop-clearance, but it isn't just the length of the nose wheel that is affected. The main gear can be shorter on a tailwheel aeroplane because the fuselage is tilted backwards when on the ground raising the prop. This saves even more weight and serves to reduce drag. Another issue with a nose wheel is that a tail-strike can cause damage.
Surely the only real benefit to a nose wheel aeroplane is easier ground handling?