If clumsy is used to describe a person, it usually means that that person lacks physical coordination: they trip and fall over a lot, or are always dropping things, or something similar.
However, clumsy can also be used somewhat metaphorically to mean "disjointed" or "haphazard" or even, as dictionary.com puts it, "ill-contrived." Example:
Donald Trump's tweet of himself eating a taco salad was a clumsy attempt at connecting with Hispanic voters.
Awkward, when used to describe a situation, can be pretty similar to the metaphorical use of clumsy, or it can be used to mean that something about the situation was socially weird: like if you're trying to make a good impression with your significant other's parents and you make some offhand comment about "oh I hate pie, I prefer cake," but then you find out that one of your SO's parents owns a bakery where they only make pies.
When awkward is used to describe a person, it usually means they are socially awkward: maybe they ask questions that are a little too personal when first meeting someone, or they have a habit of standing too close to people, or whatever.
Mostly, I would say using "clumsy" and "awkward" in the same sentence would be overkill and possibly redundant, but maybe not necessarily. In your example sentence
He tried to dance, but he was too clumsy and awkward.
I would say that clumsy describes the physical lack of coordination that prevented his dancing from being good, and awkward describes the unpleasant atmosphere that resulted from having to watch his awful dancing.