It's a cup is half empty, half full kind of thing.
One says, life is full of suffering so one succeeds by coming out a winner ahead of others.
The other says the life is full of suffering, so one must help everyone out of suffering to personally come out of suffering.
One view point creates conflict, the other eliminates conflict.
I see Art of War as short term thinking, and Buddhism as long term thinking. The former has instant appeal, but the latter has appeal to those who are contemplative and not hasty.
Note: I'd like to mention the short book (50 odd pages), "The Buddha Taught Nonviolence, Not Pacifism" by Paul R. Fleischman, MD on the subject of violence. It's a difficult subject, I can't say I agree with the entire book, but it has interesting ideas.