@BellaB makes several good points. There are many Quebec dialects, just as there are many French dialects. However, Joual is a very rough, rural dialect, and a very specific form of Québécois. You don’t hear Joual in the streets of Montreal; even for a French speaker, it’s extremely difficult to understand. I think there is often a tendency to label all Québécois as Joual, which is incorrect.
I think the main differences between French and Québécois are that Québécois is quite slangy, and I find that the mouth is a bit more relaxed. When I’m speaking French in a professional capacity, I tend to tighten my mouth, speaking more from the front of the mouth, and I change my vocabulary quite a bit. I went to English (i.e., English-language Canadian) schools in Quebec as a child, and the French we learned was called “Parisian French”, but I rather doubt it has any relation to French actually spoken in Paris (just to make things confusing).
I’ve posted this video on Fluther before; it’s by a French woman who grew up in Quebec, explaining Québécois vocabulary to people from France, so you can hear her use both. She talks about some of the peculiarities of Québécois. For example, in Québécois, you tend to add an apparently unneeded “tu” in some contexts; there many, many more contractions that don’t exist in French; and the usage of some words is quite different. But it’s also important to remember that the language she’s speaking is very informal. This is how you’d speak among friends; in a shop you’d be more formal.
There are also words in the Québécois vocabulary that come from our history. So, for example, we lock the door in Quebec by saying “Je barre la porte”. In French, “Je verrouille la porte”. This is because, in early Quebec, the “lock” was literally a wooden bar that would slide down to lock the door. There are a few odd words like this.
So, essentially, there are wide differences between Québécois and French in an informal setting. In a formal setting, the language is very similar, although there would be differences in accent, which I think mostly come from how we hold ourselves when we talk. Again, that mouth position thing.
I hope @berserker drops by, because she’s a native Québécois speaker, and her perspective will undoubtedly differ from mine.