You should try both. These are different so it's not really an either/or situation. It depends on what sort of "shopping experience" that you want (yuck-speak, but I can't think of anything better).
Cabot Circus is an extension of the City Centre Broadmead shopping area. This is the new shopping precinct and is quite stunning to look at (well designed city centre facilities are rare). It contains pretty well everything that you would expect to find in a big High Street as well as large department stores, but all rolled up into a multi-story precinct. In the Broadmead part there is an excellent Debenhams. There is also a very good selection of eating places, from very cheapo to cordon-bleu. You are on the edge of Bristol city Centre, with all the great things that are there: The Matthew, The SS Great Britain etc.
It is a 10 minute walk from Temple Meads station (5 if you're fit). There are buses if you're laden with shopping bags.
Cribbs Causeway is the out of town shopping centre. It is enormous. People come from beyond Exeter to spend the day there. The highlight of the area is "The Mall". This is a long two-story precinct with a huge Marks & Spencer at one end and a massive John Lewis at the other. Surrounding The Mall is a bunch of superstores: Asda/Walmart, PW World, Curry's, Morrison's, etc. There is a big cinema multiplex, gym and other entertainment things. Eating is easy with a choice of feeing houses ranging from Harvester (on the edge and you have to cross a main road) to TGI Friday's. There is not the haute cuisine that exists in Cabot Circus.
The nearest main line railway station is Bristol Parkway, but you'll have to catch one of the connecting buses (about 20 mins ride) - it's too far to walk. There are buses from the Centre, but it's a long slow journey.
If you decide to travel by car - Cribbs Causeway has acres of free parking. Cabot Circus has lots of multi-story car parks but you have to pay and navigate the city centre traffic to get to them.