The main difference is geography.
Chevy Chase is a swanky just-barely-suburb of DC to the north with lots of expensive shopping, restaurants and some nightlife. Right on the Red line metro, it's about 20 minutes from downtown and the Mall.
Alexandria is a swanky just-barely-suburb of DC to the south with lots of expensive shopping, restaurants and some nightlife. Right on the Yellow/Blue line metro, it's about 10 minutes from downtown and the Mall.
I'd stay in Alexandria because MARYLAND SUCKS. (sorry little regional prejudice)