I believe that either is correct, as others have stated, but I sense a slight difference in connotation between the two phrases.
Competing against someone or something indicates a targeted approach to competition where a competitor is focused specifically on one of its competition. I sense a bit of aggression or animosity in the phrase.
Competing with someone or something, on the other hand, seems to imply a less targeted, more individualist, less antagonistic form of competition.
Apple competes against Google. I compete with others in the labor market.