A drying agent simply absorbs the moisture or water vapour that is present as an impurity or contaminant with the desired substance. It does not act as a reactant to produce a new substance by a chemical reaction, but simply as a purifier.
For example. Concentrated H2SO4 can be used as a drying agent to remove any moisture that is present as impurity with gases like CO2, nitrogen or oxygen.
Anhydrous CaCl2 is another good drying agent used in chemical analysis.
A dehydrating agent, on the other hand, actually acts as a reactant. It helps formation of a new substance by removing some parts of the other reactant as water. It effectively decomposes the other reactant by breaking chemical bonds.
For example, phosphorus pentoxide dehydrates nitric acid to dinitrogen pentoxide when it is distilled with the latter.
Alumina (Al2O3) and conc.H2SO4 are also widely used as dehydrating agents, especially in converting alcohols to alkenes.