A touch up describes the process of "touching up" the new growth to match the rest of the relaxed hair. Generally stylists call this a relaxer touch up.
The relaxer is the product used. Now, a "full" relaxer denotes having all virgin hair relaxed for the first time. A touch up is done on roots.
I think these are the same...maybe if someone is referring to a virgin relaxer (i.e. relaxing the whole head) then they'll say relaxer as oppose to touch-up, but for ladies who already have relaxers, I think they use "get a relaxer" and "get a touch-up" interchangably. They both refer to relaxing the new growth, I think.