In which contexts the usage vary?
Though they both imply there is some kind of mistake, they are pretty different kinds of mistakes.
a fallacy is an error in inference. You have some facts and do some reasoning but come to a faulty (or unsubstantiated) conclusion. Then you've committed a 'fallacy'. Coming to a wrong conclusion is a fallacy.
a misnomer is simply a wrong name for something, in the sense that the given accepted name is at odds with the nature of the things itself. A 6'7" basketball player named 'Shorty' has a misnomer (the name 'Shorty' is a misnomer).
A fallacy is a common misconception. A logical fallacy is one type of example.
Misnomer refers, more specifically, to a wrong name or inappropriate designation -- for instance, calling Native Americans "Indians."
An easy way to remember the difference is to remember that misnomer is derived from the the Latin word for "name," nominare.
I think Wikipedia has one of the better succinct definitions: "...a fallacy is incorrect reasoning in argumentation resulting in a misconception." It should be noted that there are a number of distinct logical fallacies, and that an entire section of philosophy is devoted to them. The referenced link is a good start for exploring the various flavors of fallacies.
@Mitch's answer to a misnomer is well stated.
A fallacy has a technical and a common usage, but both could be loosely summarized as "an error in reasoning" or a "logical fault".
Poor reasoning can also be called fallacious reasoning.
"Begging the question" is an example of a fallacy. From nobelief's fallacy list:
begging the question (or assuming the answer): (e.g., We must encourage our youth to worship God to instill moral behavior.) But does religion and worship actually produce moral behavior?
Here's an example usage from a comment on Serendipity: Or, What has Software Engineering got to do with Climate Change?
I only bother to bring it up because, IMHO, an appreciation of this fallacy is necessary to appreciate the real differences between technical software validation and its verification.
Something being a 'fallacy' tends to indicate an error of process.
A misnomer is simply misnaming something, or calling something by the wrong name or a misleading name.
This can be minute and technical or broad and ideological or philosophical:
Calling that little hill Mount Brawn is an egregious misnomer.
Calling that a budget plan is a misnomer -- there's no plan, it's just budget fraud.
It is incorrect to use misnomer to mean an incorrect assumption, poor reasoning, or more general kind of error where another word should be used instead:
Don't use it this way, please:
It's a misnomer that all dogs bark.
Ugh. From the another comment found in the wilds of the Internet:
Again, please, no:
There's still a misnomer that single women pushing thirty are fatally flawed or intimidating[...]