The Alphas were genetically engineered to fight against Alpha Quadrant species. They believe themselves to be superior to the "Gammas", Jem'Hadar who were bred in the Gamma Quadrant, leading to considerable friction between the two groups. They believed that their actions better demonstrated their loyalty to the Founders than just words. They were also more willing to take independent actions which advanced their goals, while Gammas only did what they were ordered to, and no more.
The first significant combat action of the Alphas was the brief capture of the USS Defiant, under Kudak'Etan. (DS9: "One Little Ship") In addition to being specifically engineered for combat against Alpha Quadrant species, Alphas had a slightly different skin tone than their Gamma counter-parts, appearing to be less blue-ish grey, as well as head-crests in place of the gamma's bone-mohawk ridge.