They're totally different beasts. Street riding is about being fluid, dirt riding is more...sporadic.
As stated, (imho) dirt riding is just safer. Buddy of mine got hit by a soccer mom (who was in his lane no less) driving her SUV, talking on her cell phone, and yelling at her kid in the back seat. Lost an arm...
Trees and rocks won't be in your lane. If you get hurt 99.9% of the time it's something stupid you did. If you get hurt on the street sure, you mighta done it, but there's a good chance it was someone else's stupidity that you'll be paying for as well.
I'm looking to give up the street and just stick to the track, and then get a dirt bike. Best of both worlds :)