The Speed Boost is a boost applied to your robot's Base Speed by adding movements parts, particularly Thrusters and Propellers. Note that the weight and proportions of a robot will effect the Base Speed.
The Damage Boost is a boost applied to your Base Damage, which means the damage attribute of all weapons on your bot, depending on your robot's CPU. Robots with lower CPU will experience damage boost, but robots of 2000 CPU and above will not have a damage boost.
Damage Boost Graph
The three types of boosts were added to Robocraft one by one during beta test (02/03/2017 - 24/08/2017), leading up to version 1.0).