First, let's assume away the most likely outcome: Massive resistance by professional drivers will kill the adaptation of the technology. Why take this step? Because if the technology is not adapted then there is nothing to speculate about and that is boring.
1. It will be less common to own your car. The main reason people own their car is that they want to have one available. Even if you wanted to share your car, it is not always convenient to do this. A person could use my car for the ten hours I am at work or the 13 hours I am at home on a normal work-day, but they would have to live near one of those places and have to restrict their use to those windows of time. But what if my car could drive itself? It could go and pick-up someone who works later than I do or drive around a mom who has some mid-day errands to do.
There is a lot of trouble and expense in owning and maintaining a car. All of that cost provides me with an hour per day of use. For only a marginal increase in cost, the car could be kept busy all day long. A company can operate and maintain a fleet of cars for less per car than individuals could do. If they can do this and make sure that subscribers have a car when they need one, why would people bother to own one? Some people would remain attached to an old car or one which is special to them in some way but a lot of the fun of driving will be gone because:
2. It will become illegal, or very expensive to manually drive. The software and sensors are improving all the time and it won't be long before computers are much, much better at it than any human. The temptation will be to outlaw the risky behavioral of manual driving. My kids, or maybe theirs, will be the last generation to know how to drive.
3. Long range travel by car will be much more tolerable and therefore more common. Computers would allow for faster driving and could save fuel by networking with other cars on the road to tailgate each other and thus save on wind resistance.