By the power of truth, I, while living, have conquered the universe.
Halo 2 was rushed by Microsoft's strict release dates, so the Brute Plasma Rifle was added to it in place of the Spiker. Halo 3 had much more development time to work on the sandbox, they were able to unleash the Brute weapons.
Also, from a canonical point of view, the Brutes weren't in power until the end of Halo 2 when the Great Schism begins. By Halo 3 they'd have been allowed to remove the Edicts the Sangheili placed upon them, so they could use their full military might.