What activities do you want to do while in the city?

Mayflower: located in the western part of downtown (south of the Dupont Circle neighborhood). This is in the middle of a commercial area, so will be quieter on weekends. Walking north will get you a bunch of restaurants and some museums, and it's also a quick metro ride to the zoo. A a few museums are relatively close (the Philips Collection is the most notable one IMO). Not a far walk to the National Mall; directly south of the hotel on the Mall are the monuments. The museums will be further east.

Downtown Renaissance: located near the convention center and Verizon Center/Chinatown. Tons of restaurants and some museums nearby, and it's under a mile to many other museums on the National Mall. A more lively area, even on the weekends as there's a mix of office and residential in this part of town.

Both hotels have easy access to the metro (Dupont Circle and Gallery Place stations, respectively).