To me, section implies some formal division between areas of an object or place, whereas part is a more general term which doesn't have a clear boundary. Neither say anything about size.
Both of your examples are grammatically correct, but the word choice gives you some nuance. So, "the front section of the car was damaged", would suggest that the front area of the car, up to some structural division was damaged.
Your other example:
In sections of Canada, French is the first language.
This sounds a little odd to me. In parts of Canada... sounds more natural, unless you're trying to highlight some boundary between the areas.
Below are some other examples with the more natural word (though either word would be strictly grammatical). Note how section highlights formal divisions, whereas part is used for things without clear boundaries:
The violin section of the orchestra.
Which section is non-fiction in?
The first section of the ship was assembled on time.
I'd like that part of the cake.
I like to sit in the warmest part of the room.
Parts of the country are suffering drought.
Edit: Note that sections tend to be made up of smaller individual pieces. Violinists in an orchestra, books in a library, or words in a newspaper (as in Aaron's example).