Fragmentation occurs in multi-cellular organisms with relatively simple body organisation. Regeneration on the other hand occurs in fully differentiated organisms.
In fragmentation, organisms simply break up into many pieces upon maturation and each fragment grows into a new individuals. E.g.: Spirogyra
In regeneration, if the individual is somehow cut or broken up into many pieces, many of these pieces grow into separate individuals. For example, simple animals like Hydra and Planaria show regeneration.
Regeneration is carried out by specialized cells, which proliferate and make a mass of cells from which different cells undergo changes to become various cell types and tissues.