Regeneration is........an incredibly convenient way to replace the lead actor.
OK, kidding aside, I've always thought it had something to do the "time energy"/Time Lord power/whatever within every Time Lord. When he's "killed," the energy escapes, causing the dying cells to regenerate and mutate slightly/alter the DNA in some fashion. So, he's physically different, but also slightly chemically different, so his personality changes as well. Or something. Wibbley-wobbley, incredibly convenient, time-whimey stuff.