Two things, actually:
- They don't show up on radar. So your local weather coverage can't tell you exactly where the tornado is. That is one reason why they tell people across a wide area to take cover: tornadoes move very fast and in erratic paths, and even with radar and satellite they can't say precisely what or where is in danger.
- They tend to TRAIL storms. This is very important. If you're watching local weather during a tornado emergency and the red center of the storm has already passed you, you still may be in danger.
I learned these things in 1998 when a tornado passed just to the north of me. I had been watching weather TV, watching the Big Red Dot at the center of the storm, and it was well on the way to Wilmington (where, ironically, the National Weather Service is) and past me.
Then the power went out. When it came back on in 45 minutes, the TV was reporting fatalities just up the road from me, a mere mile away. (For reference, a tornado can cover a mile in less than 60 seconds).
I had been, no joke, extremely lucky.