The Curtius rearrangement and the Hofmann rearrangement are two important reactions in organic chemistry that involve the migration of different types of groups.

The Curtius rearrangement, named after its discoverer Theodor Curtius, is a process in which an acyl azide is heated to produce an isocyanate. During this reaction, an acyl group ($R-CO-$) migrates from nitrogen to oxygen. The resulting isocyanate can subsequently react with various nucleophiles to form amines, ureas, or carbamates. This reaction is especially valuable for synthesizing amines and other nitrogen-containing compounds.

Conversely, the Hofmann rearrangement, named after August Wilhelm von Hofmann, transforms an amide into an amine with one fewer carbon atom. This rearrangement involves the migration of an alkyl group ($R-$) from carbon to nitrogen. The reaction commences with the oxidation of the nitrogen in the amide, yielding a nitrene intermediate. This intermediate then rearranges to produce an isocyanate, which can be hydrolyzed to form an amine.

In summary, both the Curtius and Hofmann rearrangements involve the migration of a group from one atom to another, but the key distinction lies in the type of group that migrates. In the Curtius rearrangement, the migrating group is an acyl group, whereas in the Hofmann rearrangement, it is an alkyl group. Additionally, the Curtius rearrangement is typically employed to synthesize amines and other nitrogen-containing compounds, while the Hofmann rearrangement is used to convert amides into amines with one fewer carbon atom.