i've been on both sides of the fence on this one, as a street racer in my teens and then as a track racer afterwards. The main difference between both boils down to one simple word......maturity.
See, the people who mostly street race are usually in their late teens or early 20's, so they are immature, think that they are above the law (i did back then) and don't care about who else is around, so long as they are able to get the win. When you combined nearly 1000 cars and 5000 people (which was quite regular in the mid-90's in sydney) you were always destined for trouble. You get people doing stupid stuff like donuts and burnouts, surrounded by spectators with no regard for their own safety. These things usually add up to people being injured when a car goes out of control and spears off into the crowd, or the rear end swings around too fast during donuts for the people to avoid the car and get whacked about by the car. You also have those same spectators trying to make the burnout better by lifting the rear end so there's less weight, or putting diesel on the tyres to make it spin easier. When there's a drag race, you have a person standing between two cars which can at any moment jump towards that person on launch and sometimes hitting them as they take off.
On the actual track, you will never see those things happen. Sure, there's heaps of spectators, but they are placed in more appropriate places like the stands. If there's a car on the burnout pad, you don't see 400 people surrounding it trying to assist it in it's burnout. If there's a drag race, you have things like the safety safari, ambulances incase of major accidents, prepared tracks, christmas trees, all the things to make the environment safer. Most importantly, you have the police on your side, not out there to impound your car because you're doing the right thing.
In australia, we had the police participating in legal off-street drag racing (one of the other aussie's can help me, vp commodore running 10's, can't remember the name of that car), using a highway patrol car and there are a few places around the world (examples being portland, or & vancouver, bc) where the police have a drag car in police stripes racing up the 1/4 mile. It's not that common, but there are some police forces that see the need to promote safer racing by practicing what they preach and going up the strip themselves.
But in short, it's better to go to the strip. Usually you pay $50, and get a whole night's worth of drag racing action, plus you get to go up the strip atleast half a dozen times through the night.