Clifton is traditionally the posh part of the city where all the wealthy merchants built their grand houses high on the hill away from the noise and smell of the harbourside where their trade was done. Over the past 300-600 years things have changed quite a lot, the harbourside is full of very desirable apartments and leisure boats and Clifton is a mix of posh people and students. The restaurant prices in Clifton are still high and the central area far more diverse. The city centre nightlife is really about very late night on Fridays and Saturdays and unless you are planning on being there after 11pm or hanging out of your hotel window after midnight you will probably never know it is happening.
Car parking is certainly an issue worth negotiating from the outset wherever you stay. Personally I would find 1/2 hour to catch and ride a bus each way a bit of a pain but I also remember some of my happiest holiday memories being riding on local buses with the local people. A day-rider ticket is excellent value and could take you anywhere in the city and people are quite good at helping you with directions. Virtually every bus heads in and out of the city centre like the spokes on a wheel, most run every 15-30 minuets.
In short.... personally I would go for the best deal on a hotel preferably within walking distance of the city centre. I always liked what is now the Marriot City on Castle Park... it has a cute little round pool and sauna and was the hotel of choice for visiting actors and performers so I never knew who might be joining me in the pool or sharing a lift. I don't know what is on offer at the Brigstow but it is nicely located right on the old Bristol Bridge (not the clifton suspension bridge) and very close to the centre but not on top of it. The Thistle Grand and Marriot Royal along with the Hotel Du-Vin are the top 3... and probably the most expensive... but you might get a deal.