They mean the same, but "woo" is more blatantly romantic. "Court" has a dignified sound to it that "woo" does not have.

They generally mean the same. To court a lady more explicitly explains the process of going on one or two dates or more. The male (presumably) develops a relationship over time. To woo al lady generally describes the process when a male (presumably) makes a woman feel excited instantly on first meeting her. However it is not a hard and fast rule.