I used to know someone who went to a booksigning with Anne Rice and she asked about the whole AIDS/HIV issue. According to her, Anne Rice's comment was "They're immortal. Get it?"
I agree with Quentincollins that although technically Countess Zeleska in Dracula's Daughter was the first "reluctant vampire" on film (the title character of Varney the Vampire in the early 1800s also was sometimes very relucatnt), Barnabas was the first "hit" of such an idea and almost certainly ushered in Louis, Nick, Angel, etc.
But comparing Barnabas and Angel is a bit tricky because you have to remember their vampirism was very different. Joss Whedon's shows explicitly described what a vampire was--namely, a human whose soul has been replaced by a demon. Not, interestingly enough, their mind. The mind is the same. Internal evidence of the both series would seem to indicate the soul is something like a spiritual "organ" that encourages people to feel empathy for one another as human beings, which seemed to be a foundation for moral standards. Take that away and it is like removing every single inhibition, then replace them with the drives of a predator.
So far so good (kinda/sorta).
Yet vampires of DS seem different. It seems as if becoming a vampire altered the personality, but mostly by giving the new undead an overwhelming hunger as well as more aggression. This showed up in all sorts of ways, but in Barnabas' case this meant a much more fierce temper (as poor Willie can attest). Yet unlike Buffyverse vampires, DS vampires seem to feel waves of bloodlust. Barnabas, interestingly, went through long periods without feeding (this may have to do with the sheer length of time he was undead). Angelique showed the capacity for self-control but no real interest in it, whereas vampires like Chris Jennings just seemed to be appetites with legs. One suspects it is largely a matter of willpower and personality.
On the other hand, certain aspects of Barnabas' story do seem to resonate for this kind of story. The troubled family, the love/hate relationship with one's maker, the dreadful toll taken by those close to the vampire, the struggle for some kind of redemption through good works--all these are present in Barnabas, Louis, Nick and Angel when you think about it. Details vary quite a bit, but the essentials are there and one can note how other "reluctant vampires" failed to grab the imagination--including the lead in Moonlight and that show on Lifetime--also failed to have these elements. Not that I'm claiming that this is a formula for good vampire story-telling, but that those details did resonate and if you're gonna tell a truly rivetting tale of this genre you need to have details that resonate in some such way.