The R21 mechanism is meant to support research projects in which proof-of-principle has not yet been established. The aim of such an award is thus to acquire, through a series of quantitative milestone completion, sufficient data to demonstrate proof of principle.
The R33 mechanism is meant to support research projects for which proof-of-principle has already been established. This mechanism is more developmental in nature and thus serves as a continuation of an R21 type of award. This mechanism is meant to support the further maturation of a technology over a 3-year period.
Specific requirements for each of these types of grants are outlined in the funding overview section of this website and in each of the relevant RFAs.