The radar can't distinguish between overhead signs, trees and things on the side of road. It only track moving things, otherwise there would be too many false positives and the car will keep braking every few minutes. Other Level 2 autonomous systems have the same limitation.
You might recall, Tesla hit a stationary fire truck in Jan [0]. This was caused due to a similar limitation in their software. Tesla was following a car at a slow speed. When that car suddenly moved out of the way to avoid hitting the Fire truck, Tesla AutoPilot accelerated to match the set speed and hit the fire truck. To summarize, if a moving car suddenly moves out of the way, Tesla will almost always hit whatever is stopped further down the lane. They even call this out in their User Manual.
Tesla is hoping that it would be able to get it's Vision capabilities to the point where it would be able figure out obstructions using cameras working in conjunction with the radar. But there is no committed timeline for this.
LIDAR is able to avoid this by using directed LASERs to build a 3d outline of all the surrounding objects. A system relying on LIDAR would have challenges in RAIN and falling snow. However under natural conditions (where most Tesla deaths with Autopilot have occurred - hitting a SEMI from side, hitting the barrier recently), LIDAR would be able to completely avoid the accidents.
However, if and when, Tesla vision capabilities are able to detect obstructions, they might be much cheaper and more effective than LIDAR in all kind of weather condition. Hoping they can get there soon. And I also wish they would fix their marketing to more accurately reflect the capabilities of the driver assist system and stop misleading the public.
[0] https://www.mercurynews.com/2018/01/22/tesla-on-autopilot-sl...