Hydrogen peroxide is a stronger, faster bleaching agent than carbamide peroxide, that's why it's usually recommended for use only by dentists. Carbamide Peroxide is a lower-dosage formulation of hydrogen peroxide that's safe for at-home use - or for use over a longer period of time, like with whitening kits. Most over-the-counter whitening products use carbamide peroxide while professional whitening treatments use hydrogen peroxide.
Carbamide peroxide breaks down to about one third hydrogen peroxide, so an 18% carbamide peroxide solution has the same effect as a 6% hydrogen peroxide solution. Another reason carbamide peroxide is used in at-home whitening products is because it is much more stable than hydrogen peroxide, which deteriorates rapidly on store/dentist shelves.