That all stories require some sort of conflict might just be the least controversial thing I've ever said in my life. In fact, I’m pretty sure one can define a story more by its conflict than anything else within it.. This can be anything from a man vs. man to man vs. self to man vs. society to, well, anything. So long as you have two entities (no matter how you define those them) who want different things, you have a conflict; and therefore, a story.
One of the strengths of Vampire is the wide variety of Conflicts it can encompass. You can go with the straight up, rock ‘em-sock ‘em violence of Camarilla vs. Sabbat, the subtle paranoia of Diaberlist vs. Targets, or the existential conflict of Beast vs. Humanity. In a way, Vampire is a “kitchen sink” game when it comes to Conflict. Pretty much any struggle you can imagine, from the most gritty and realistic (Beast vs. Humanity) to the most outlandish and comic booky (Tal'mahe'ra vs. Aliens) can be expressed in the game.
Personally, I like Vampire vs. Vampire. Despite my affection for Humanity, the quieter, subtler struggle of the Beast isn’t something that really wows me-though I’d love to play in a game that proves me wrong. Same goes for Vampires vs. Humans in a Lords of the City game. While these kinds of conflicts can make for interesting and fun stories on occasion, when it comes right down to it, nothing makes for a more exciting tale than vampires feuding with other vampires.
A lot of this is due to the Pax Vampiricus that’s in effect in most of the sects. The Camarilla has the Tradition of Destruction, while the Sabbat keeps the direct violence in check with the widespread use of the vinculum. As such, direct, violent action is forced onto the back burner, necessitating more complex and interesting tactics to thwart one's foes. Not that violence and combat can’t break out, of course, but it’s something rare, and only engaged in with some element of trepidation.
This is what makes Vampire fairly unique and unusual for me. I haven’t seen many other games where violence isn’t a viable solution to most problems. Obviously, games like D&D thrive on combat, and killing your foes is often not only a solution, but often the best solution. Even games that try to tamper down on the violence do so by either introducing lethal combat (and all the mechanical complicated that entails) or by creating monsters that are so terrifying that mere guns and blades can not hope to harm them--the game is instead more of a “puzzle” to be solved.
Vampire is different. You get to play supernaturally powerful killers, who have every reason NOT to use your supernatural powers to kill. It’s an odd contrast, but that tension is a key reason why Vampire works so well. Each character is perpetually one step away from lashing out, and the one that does is often the loser. By struggling against others that are so encumbered, you allow for far more interesting and complicated stories, and great deal more work and effort to actually resolve these conflicts.
It’s not the same with other supernaturals. Sure, a coterie of vampires will need to approach a Werewolf or a Mage with a great deal of trepidation, and only strike when the odds are in their favor. After all, once can turn you into a lawn chair with a look, and the other...well, they’ll have to spend a round clawing you all to death, then use their Crafts skill to turn you into a lawn chair. The point is, if you screw up against one of these guys, you’re gonna have a bad time. But, they’re still conflicts that can be, and often are, resolved with death and violence.
Which makes them fun, for the occasional story. It’s enjoyable to roll some dice and get your fangs dirty. But, killing Werewolves, no matter how many different tactics one uses, lacks the nuance and depth than comes from going up against a fellow Jyhader.