It's true that everyone's definition is different, so I'll give mine for the sake of research. I'm typing this up without reading it over too much, so if I offend anyone, I apologize in advance - this wasn't meant to be degrading to one group or the other in any way.
I think furries are people who like the art, fursuits, whatever. Just people that like anthropomorphics in some degree (some of course more than others). Furries can choose their fursona, they could make something up because they think "Hey, I've never seena platypus-anthro before" and make it their fursona if they want. Or they can feel they have a certain connection to an animal, usually not to the degree of being trapped in the wrong body. Some don't have a fursona at all, they could be here for the art or the community.
Weres are in my opinion the pople that feel they don't just have a connection to some kind of animal, but that they have an animal's soul in a human body, sometimes they share traits with their animal. The majority of weres believe in some kind of shifting - mentally or physically (even though no one's actually proven that). Some grow phantom limbs belonging to their animal side. As far as I know (granted, that's not much further than I can toss a car) not all people have an animal inside them waiting to come out, like weres have. This may make some weres more elitist than furs are.
You could be a were, but not be a fur.
You could be a fur, but not be a were.
You could be neither or you could be both.
It's all possible.
If you think that you're a an animal's soul in a human body, but you don't agree with what other weres tell you (about furry, about the nature of stuff) that doesn't change who you are. If you're a were, you can be a were without being in the community if you don't like the people there.
Again, the above is just my take on things, and it's probably not even what you'd call well-written. It probably doesn't reflect my exact feelings on the matter, but I can't put it into words any better than this, sorry. It's probably not even on-topic anymore.