The demo won't have any part of the "story" in it. People always say Monster Hunter doesn't have any story to it and that it doesn't need one. Tri was my first MH game and I was actually pleasantly surprised by the story it did have. It was fun, cute, flirty, funny, macho, heart warming, etc. It's not going to win any awards for the story it tells, but to say it doesn't have a story doesn't make sense to me.
I guess I never understood why people always say it doesn't have a story when it had a nice one in my opinion. If you just play online then there is no story, you just meet up with other players and hunt and craft, etc. But the offline single player definitely has a story. It has plenty of characters, it has emotion, it has a plot, it has a beginning, middle, a climax, and conclusion.