The reason why their stormtroopers are equipped with different armour types is due to the various environments and war theatres the Empire's armies operate in. The normal Stormtroopers are the generic Empire shocktroopers . There are Sandtroopers (Tatooine environs), Snowtroopers (Hoth environ) bikerscouts(Endor environs), Zero-G troopers(Space environs) and many more.
The Republic Commandos are like the Green Berets or our Commandos, unlike the normal clone troopers. They are specially trained for sensitive and dangerous missions. Various units were involved in different operation theatres during the Clone War period against the Separatist factions.
It must be noted again that they are operating under the banner of the Old Republic and not the Empire yet(In the game). I assume that when Palpatine declares himself Emperor, their allegiance changes accordingly, as does the thousands of Clone Troopers. The change in their loyalty/allegiance also corresponds to their change in their armour; to that of the Stormtrooper armour that we are all so familiar with.
May the force be with you.......
