Road switchers were designed to bridge the gap between a yard switcher and a "road" unit. Road units have larger high horsepower engines and more elaborate trucks designed for higher speeds. The number of axles has more to do with the need for additional traction motors or a need to distribute weight over more axles to handle lighter weight tracks.
The first succesful road switcher was the ALCO RS 1. It pretty much set the pattern foreverything that followed. Typical early road units were "cab" designs. These were build up from stamped steel u channels welded together in truss structures to make a bridge. The deck of the bridge was not a frame per se, the load was carried by the side trusses. These provided good ride and great visibilty only at the front of the engine and no place for brakeman or any other crew. You couldn't see much going on behind you.
Yard switchers had small engines and built up box beam frames. Great visibility but very poor protection. They were not designed for speed or distance work.
A road switcher has a frame like a yard switcher, albeit longer, a large engine like a road unit, and road trucks. It can be used for road work but can also be used as a switcher. As the hp increased and the distance the engines were expected to cover increased the cabs were upgraded to allow the crew some additional comforts (including a potty).
Full width cowl units are basically road switchers with a full width metal cowling. These are usually found on roads were the outside environment makes the typical narrow hood less attractive. The crew can do minor repairs without going outside or move from unit to unit with less exposure to wind/rain/snow/sand.