There are various different explanations across universes. But the biggest issue is - why can't vampires just drink animal blood? It has to either be some kind of magic, allergy or inefficiency usually. Regis exists to show the book's theme - that not every monster is a monster, and that not every human is human. So he needs a reason not to hurt people, otherwise he's also a monster.
Does it make them weaker or do they just die because they need more nutrients?
I remember Lumley's Necroscope having some spiel about drinking vampire blood too. But that series of books has vampires be a leech-like creature that infects a host, not an illness or something magic that does that
Well VTR is a roleplaying game. It's similar to Vampire the Masquerade, but different setting and somewhat different mechanics. I guess it's best explained as "nutrients." Animal blood and blood from e.g. blood bags gives less Vitae (magic blood points resource) than blood harvested from living humans. And as the character becomes more powerful, eventually that "lesser" blood can't actually give them Vitae.
The vampiric curse in VTR is explicitly stated to be supernatural, though, so there's not a necessary scientific explanation for it. The curse imparts the Beast, which is the predator in all vampires.
In BTVS they can. There's a vampire who's cursed with having a soul and is therefore unwilling to hurt humans, so he just toddles down to the butcher and gets blood from them.
Soulless vampires either like the hunt or just prefer the taste of human blood. For instance, some vampires just raid blood banks instead of hunting humans.
I like vampires in the Pratchett universe, who often see human hunting as a sport and leave religious symbols, stakes, and garlic around the castle to level the playing field. (Not to mention the black ribbon society, the vampire equivalent to AA.)