This is a really good thread and I don't think I can add anything that hasn't already been said. I'm a professional storyteller and folk singer, one of the few who work on both circuits plus with the general public who wouldn't venture into either world. Other people have said it but I'll stress it: tell a story you like and respect and tell it with conviction and tell it to the audience which is there in front of you - reacting to their reactions. It will come out slightly differently every time depending on whether they are adults in a story club or adults in a folk club or a mixed family audience or all kids. But if you are honest and doing it properly they will all appreciate it in their own ways and will get different things from the story. Although there are some stories you wouldn't tell to children and some obviously not meant for adults. Some kind of scene setting or explanation of why you're telling that story at that time can help - unless the reason is that it's the only one you know!
A couple of people have mentioned style and I think that's what puts off some folk club audiences. They don't like the telling to be too actorish, it must be natural and honest (I keep coming back to that word) in the same way that song presentation must be. It's the story/song that counts not the cleverness of the artist who is just the transmiter.
For more info. background and ideas you could try Facts & Fiction storytelling magazine (which I happen to edit!) Look at www.factsandfiction.co.uk