While you are waiting for the guys who can explain these things to answer, read this thread. By the time you are done, hopefully a knowledgeable person will have a nice succinct answer regarding suspension, gear ratios, cooling systems, maneuverability, overall power. The thread below will give you an overview of the arguments in a "yeah but what about?" fashion. (In the thread the R bike is the mx bike, the X bike is trail.)
The MX bikes are all about race. They have the highest performance engines and are the lightest weight. A serious racer wouldn't want any extra weight. Many race organizations don't even allow kickstands on the race bike for fear of them coming loose and becoming a hazard.
Some of the MX bikes have an off road cousin, but not all. Take Yamaha, for example. The YZ line has two strokes and four strokes. The YZ-125 and YZ-250 two strokes have no "off road" equivalent bike. The YZ-250F and YZ-450F have a close cousin in the WR250F and WR450F. Both models have essentially the same suspension and the basic specifications for the engine are the same. The gear ratios in the transmission may be significantly different, the engine is tuned slightly differently to meet the EPA specifications, the addition of the lights, battery and starter are significant, and they include a kickstand.
The Yamaha line of "off road" bikes that have no MX equivalent is the TTR line. These bikes are all four strokes, use a smaller frame (better for shorter riders), have significantly less suspension travel, and the engines are a much lower compression ratio (allows them to run on regular gas). They won't produce as much horsepower but they should last forever.
rmc_olderthandirt, back in the day the YZ125 and 250 both had off road modles wich were the IT125 and IT250. The in the early to mid 90s the YZ250 had a WR250 brother. But as of noe those days have past and now it all about the 4 stroke. Not that there is anything wrong with that because each to there own.