What are the differences between Canada's French and France's French? Should I study a little bit of both? Are there some words or expressions from one that might not be understood by the other?
What are the differences between Canada's French and France's French?
Written and formal types of French are very similar. There are only a few cosmetic and vocabulary differences that do not prevent mutual comprehension.
Spoken French can be an issue. Colloquial conversations between French Canadians (see joual) might be unintelligible to France native speakers and sometimes subtitles are used in Quebec movies available in France. I suspect the opposite might be less of an issue because Canadian French-speakers have probably more exposure to French from France, its accents and expressions than the other way around.
On the other hand, I have never had any real issue understanding and being understood when meeting French Canadians, only occasional unexpected vocabulary, gender, familiarity (tu vs vous) and expressions issues that are quickly sorted out.
Should I study a little bit of both?
That's up to you.
Are there some words or expressions from one that might not be understood by the other?
Yes, probably a similar level of misunderstanding that would occur between US and UK English speakers.
Since the differences, or lack thereof, have been thoroughly covered in other answers and comments, I'd like to focus on your middle question, namely whether your should study both.
Ultimately, only you can really answer that question, but here are things I'd consider if I were in your shoes:
If your answers boil down to something along the lines of a very nebulous "because it sounds cool, but I don't really know", then you're probably better off learning "international French", i.e. the basic set of vocabulary, syntax and grammar that is share by all varieties of French. You can then gradually pick up on more specific regional diversions as the need arises.
On the other hand, if the questions lead you to a very specific answer such as "I want to move to Montreal and work in the _____ industry", then you'll want to focus on studying the French Canadian variety of the language.
If you're still unsure, you probably can't go wrong if you simply start by getting a firm grounding in the basics of the language. You'll be able to build on that to adapt to any more specific needs you might have later.
The differences aren't that big. The major gap would be the accent and some of the expressions. There isn't two different languages to learn, that's just one, with some variations. It's just like British English and American English. You just need to learn French, in general, to be able to understand Canadian French, and general one.