Dana,
I made that trip in June and would recommend using the AeroBus, as it will be the cheapest method. I don't know what a Taxi would cost from the airport to Plaça Catalunya, but likely at least €50 or more if the traffic is thick. I found that in some parts of that trip, traffic was "stop & go" so I was glad I had chosen the Bus.
Luggage shouldn't be a problem. As I recall, there's a rack in the centre of the Bus where you can place larger items. Depending on how crowded the Bus is at the time, you should be able to keep carry-on and smaller items with you. I doubt there are any restrictions on the number of bags (within reason of course).
"Does the taxi take you directly to your hotel or does it drop you off and you have to walk."
That will depend on which hotel you're staying in. The hotel I was using was part way down La Rambla (which is primarily a pedestrian zone) and located in a small side street, so I had a short walk in any case. I doubt that a Taxi would have been able to access that particular hotel so a walk from Plaça Catalunya would be necessary. Other hotels would have easier vehicle access.
Happy travels!