Aim speed affects the speed to switch from aiming down the sights to the hip and vice versa.
The movement speed only takes effect when aiming down the weapon's sights.
The Greed pistol and the Interfacer shotgun prevent movement loss when aiming. With several points in Expertise, Axton can actually move faster than sprint speed while using these weapons. This is especially pronounced with a maxed out Expertise with the Slayer of Terramorphous or Legendary Soldier mods.