What is the difference between a motocross and a dirt bike as seem classified in different categories?

That is kind of a tricky question. Dirt bikes encompass any bike that is ridden off road plus what are known as dual sport bikes. Most of the manufacturers categorize their dirt bikes as Motocross or off road. Basically "dirt bikes" break down like this; 1) Motocross race bikes and woods race bikes. These are both liquid-cooled bikes with aluminum frames and racing suspension (compression and rebound settings, long travel, etc.), the woods race bikes just have some minor changes such as a wider gear ratio, a heavier fly wheel, bigger tank, lights, softer spring rate, etc. 2) Trail/play bikes. These bikes have air-cooled engines, steel frames and much less suspension than the race bikes. 3) Dual Sport bikes. This category is everything from Woods race bikes with a dual sport kit installed to make them street legal to production dual sport bikes such as the Honda XR650L and BMW 650GS.