Difference between Compete against someone and Compete with someone

compete against someone—enter into a contest with smb.:

• The male riders are determined not to let the so-called jockettes compete against them.

compete with someone—

1. = compete against someone:

• We are going to play and compete with them and, hopefully, we’ll come out a winner.

2. endeavor to hold an equal place with a person:

• These stores compete with shopkeepers, and induce them to lower their prices.