Differentials can take a little while to get your head around, but once you get the basics it all starts to make sense. I'll try and do a quick differentials 101.
The job of the differential (diff) is to let the rear wheels turn at different speeds when the car is cornering, because the outside wheel has more distance to travel (therefore turns faster) than the inside wheel.
The 3 types of diff you'll commonly come across are single spinners or "open" diffs, (probably this is what you have at the moment), limited-slip diffs (LSDs) and spools.
In normal driving, there is no difference between an open diff and an LSD, the difference is when one wheel has less traction than the other.
In an open diff, if one wheel has low traction, and the other high traction, the open diff will apply an equal amount of torque to both wheels. If one wheel can only transmit a small amount of torque because it has low traction, that will be limit the maximum amount of torque which can be sent to the wheel with good traction. The result will be the low traction wheel spinning, and the other wheel remaining stationary, or moving slowly. This is probably what happened when you got done by the cops.
With an LSD, the diff will have a limit as to the difference in wheel speed it will allow before locking the wheels together. If we have one wheel with low traction, and one with high traction, once the low traction wheel begins to spin, more torque will be transmitted to the wheel with high traction, and the car will move normally, the wheel with low traction won't spin too much. The only thing to be aware of with LSDs, is that if you apply enough power to break traction on BOTH wheels, they will both spin, whereas with an open diff, only one will spin. Many people prefer this however as it can be more predictable to deal with.
Finally, spools. Strictly speaking, a spool isn't a diff, because it doesn't let the wheels turn at different rates, it locks them together so they are always spinning at the same speed. This means when you're cornering one of the wheels is going to have to break traction to allow the car to turn. This behaviour is good for burnouts, because if you have enough power its guaranteed to spin both wheels consistently, and generally spools are much more robust than LSDs, which will wear out quite quickly if they are driven like this, but in your case won't help with keeping your wheels from spinning.
Hope that explains the difference somewhat? As Stealthy says, regardless of what diff you have, drive within your limits.