This will be my last post on the subject, and I just wanted to thank everyone for responding.
I have limited experience with neighbors, as I lived at my parents home until I was in my mid-20s, and at my new home ever since. My parents had teh same neighbors throughout those 20+ years, and they were all very easy-going. They would never object to vines infringing on their property. (They were also avid gardeners).
Likewise, my neighbors at my new home have been the same people until these new neighbors moved in four years ago. The previous neighbors - and the ones who border the other sides of my property were also very easy-going and would not object to such a thing. (In fact, one of them has vines growing on *my* chain link fence that we share, and I gladly trim them when required.)
Anyway, I guess it is my limited experience with neighbors that led me to believe that these new neighbors would not object to the vines, and made me consider them to be unreasonable in their stance.
I believe most people agreed with me that the neighbors are over-reacting, but the other side has been well-represented too. Enough people wrote in to defend my neighbors' point of view that I have to reconsider their 'crankness' as being a bit more 'normal' than I originally thought. Thanks to those who presented the opposing view, as it was a learning experience for me.
I will trim the vines for the neighbors, and if they have any more objections, I will take them down. But at that point I will reconsider my friendliness with them, and keep them at arms-length. It is still my opinion that friends should be a bit more accomodating. If they prefer to be neighbors instead of friends, so be it.
Thanks again for all your responses. I have learned a lot from all of you.
PS - You've got to admit that this topic sparked a lot of participation and some interesting debate! ;-)