These days the term quatorzain usually refers to a poem of 14 lines that the speaker does not consider a sonnet.
Sonnet is a very flexible term; you can go to the sonnet node for a list of the basic types of sonnet structure, and each of those can be modified by the poet and still be called a sonnet. While not every 14-line poem is a sonnet, the definition of what exactly is a sonnet will change from person to person.
Technically, sonnets are also quatorzains, but few people have a reason to refer to a superset of all 14-line poems. Instead, quatorzain is used as a catch-all label for all the almost-sonnets.
Sometimes quatorzain is given a much more specific definition. For example:
"A poem in fourteen rhymed iambic lines closing (as a sonnet strictly never does) with a couplet". -- the 1911 Encyclopedia. Obviously, most people do not hold this definition, as the so called Shakespearean sonnet (AKA the English sonnet), one of the most popular forms of sonnet, ends with a rhymed couplet, as do some other forms.
Or how about "anamorphic or abortive sonnets... having 14 lines and something else in common with a sonnet, typically the fact of being split in two terceets and two quatrains". -- http://digilander.libero.it/troubadours/prosody/pfor1.html .
I would not give either of these definitions too much weight; if you need a technical definition, you should get it either from your professor, or from some source more authoritative than a website.