I'm not sure how much a private driver will cost, but when you factor in the cost of the train ticket + cab fare from downtown LA to Beverly Hills, it may end up being a wash.

The Beverly Hilton is in walking distance to Beverly Hills (shops, restaurants) and Century City (mall, movie theatres), West Hollywood (nightlife), and that's pretty much it. I believe the hotel has a free shuttle for within a few miles. They are also across the street from a city bus line, with direct service to Santa Monica (i.e. the beach). From Santa Monica, you can walk to Venice Beach, which is worth checking out just because of the funkiness of the Venice Beach community.

You would definitely need a car to get from the Beverly Hilton to Hollywood (Walk of Fame, etc.), but the hotel may be able to arrange something for you. Taking the bus would require changing lines & would take far too long.