Each type of track has its own character as far as the conditions experienced by the racer. Therefore, it makes sense that each type of track would warrant a specific type of racing suit. For example, just as auto, motorcycle, and karting have their styles of suits, so do dirt and pavement tracks.
On a paved track, there are few concerns regarding the condition of the track. So long as there are no major wrecks, debris is generally at a minimum. In regards to motorcycle racers, paved tracks are especially different in that they are not constantly kicking up dirt, clay, and chunks of debris in general.
On a dirt track, however, the circumstances are reversed. All kinds of debris are flying around, making contact with you during your race. Often times, the track is also wet, presenting a completely new issue: Waterproofing.
Dirt track suits and some motorcycle/motoX suits are designed with waterproofing in mind. The last thing you want while you are racing around a track is your suit filling up with water. First of all, that would be very uncomfortable. Second, it would add a lot of weight. Third, it would raise your center of gravity. And forth, it would cause your suit (which is an investment) to deteriorate much faster. Therefore, it is easy to see why manufacturers have designed waterproof suits for dirt track racing.