Really enjoy both locations for different reasons, but would recommend County Hall. London is going to be very expensive, and County Hall has so much you can do very close by. One other feature is you can arrange to have a car pick you up at the airport, the same price as a taxi, and bill it to the County Hall. It will be a nice driver and car, much better than a taxi. The CL and Concierge's at County Hall are the best of my Marriott stays. You can email them and they will be of such great help!
Both great hotels - My personal preference is the county hall - Ive stayed there many times - If you get a Thames view room, you have one of the most iconic views from a hotel room anywhere - Big Ben, The Houses of Parliament and the London eye. The Southbank area is livley with bars and restaurants, Take a walk across the river using the footbridge near the Shell building and your close to the Strand / Leicester Square / Covent Garden areas . Waterloo underground station is only a few minutes from the back of the hotel.
I've not stayed at County Hall, but walked through it and it is lovely. I have stayed several times at Marriott Grosvenor Square (believe Grovesnor House is different) on Duke St, right around the corner from the U.S. embassy. It is very convenient--you can walk to Selfridges (and pass a wonderful Italian restaurant on the way) and the Bond St. tube station and all the shopping nearby. The area is very safe, given the proximity to the U.S. and Canadian embassies. The concierges (Harry and I believe David) are excellent and the Gordon Ramsay restaurant, although expensive, was very good. Have a great time in London!
You know, I am going to have to agree with almost all of the other posters and say County Hall. I've stayed at a few interesting Marriott Hotels in London, but the County Hall is my favourite. You can get an Oyster card to ride on the tube and it is a lot cheaper. There's also something called the London Pass which has a lot of good specials. Have a great vacation.