They're very different places. Borough Market is almost exclusively foodie, and is a great place to eat, to nibble or to buy treats to take home. However, depending on where you're headed, you may be restricted in what you can take (e.g. agricultural products). A large wheelie suitcase would be quite cumbersome here if you're visiting at the height of lunchtime on Friday or Saturday.

Covent Garden these days is much more general, it has some interesting restaurants and food stands but is much more a speciality shopping area - not as quirky as it used to be, as the chains have moved in, but there are still interesting nooks and corners and is probably a better place to spend a longer time, with more variety. However, you won't eat as well as at Borough Market!

Borough Market is best Weds-Sat daytimes (although I think some of the stalls now open Monday and Tuesday). After about 3-4ish things will wind down. Covent Garden is lively much later.

Covent Garden tube station is exit only Mon-Fri and closed completely at weekends until the end of 2015. However, other tube stations such as Leicester Square and Holborn are extremely close so that shouldn't be a factor in your decision.