The werewolves are likely to win almost every battle, but the vampires would likely win any full scale war.
Overall they aren't likely to care too much about each other. The vampires have every incentive to ignore the werewolves as long as the werewolves don't directly cause them major problems. The werewolves on the other hand do have some reason to want to eliminate the vampires because they are wyrm-tainted, but vampires are very low on the werewolves list of priorities. (A specific vampire causing problems for the werewolves is a different story...) The werewolves are likely to be far more worried about the harm to the environment, especially supernatural harm from Pentex, from the Black Spiral Dancers, from malevolent spirits from the umbra, etc. Vampires make the list, but they are low down on it.
The werewolves also have good reason to keep them low down on the list. As discussed more below, while the werewolves are likely to win every battle, they are almost certain to lose any full scale war with the vampires even on a city level.
The numbers are a big thing.
The vampires start with a huge numerical advantage and can mass produce more vampires for purposes of war in ways the wolves can't. That is to say nothing of the ghouls they can create. The vampires win any war of attrition. But its more than that.
Numbers matter in direct fights too. On average, a werewolf is far more powerful in combat than a vampire. But a group of vampires with support from a few ghouls is a very different fight for a lone werewolf.
Notably, even one-on-one, ancient vampires can be very powerful indeed compared to an average wolf.
Vampires have indirect ways of striking back.
With the right build, especially with some of the gifts available, a werewolf can have significant control over mortal society. But on average, the vampires are better at it and again have the advantage of numbers.
If the werewolves do something which will get the vampires to respond, such as randomly killing one of the Prince's favorites, the vampires can strike back in many indirect ways. They can eliminate precious kinfolk, have a cairn which isn't constantly under guard destroyed, strike at the werewolves' mortal connections, etc.
While the wolf might reasonably say they don't have moral responsibility for it, if a werewolf kills a vampire that causes a different wealthy vampire to retaliate by directly funding local Pentex's operations and lending Pentex political support then it is a major net loss for the wolves in their ultimate goals.
While unusual, it is sometimes beneficial for them to work together.
There are a lot of reasons a vampire might work with the werewolves at least temporarily and vice versa. The garou are known to see Gangrels differently from other vampires in general. Other clans have some tenuous relationships with at least some tribes as well.
Also, the wolves can advance a lot of their agenda by manipulating mortal politics, and securing vampiric aid in doing that can go a long way towards making it work. The wolves in turn have a lot to offer the vampires since targeted violence can be very useful to a vampire and werewolf blood can be valuable.
The wolves may not like working with something wyrm-tainted, but when in war, especially a losing war like the war against the Wyrm, sometimes you have to accept the lesser of the evils.