You don't NEED a car per se, but it will make your experience better. I spent one week at the hotel and did the hop-on/hop off bus the first couple of days without a car. That helped me tick off all the city sights (District 6, Camps Bay, Table Mountain, etc.). Then, I rented a car for a few days and drove around Constantia, back to Camp's bay, and to the Cape Peninsula. I thought this was a good mix of driving when it was easy/necessary and not having a car when I didn't need it.

Driving is pretty easy in Cape Town as the city is small, and the Westin is right at the end of the highway into town. Good signage everywhere too.