Let's tackle your questions one at a time:
What's the grammatical difference between these two usages of the word "awareness?"
By including a definitive article (the) before awareness, the sentence describes a specific and clearly-defined awareness that the rest of the sentence should go on to identify; in this case, the awareness is specifically of the things that he did in [the] dark. This shifts what would essentially be a complex adjective clause (...of the things that he did in [the] dark) into a noun clause by "binding" it to the noun that it is describing (awareness of the things that he did in [the] dark).
Why does one require a "the" whereas the other doesn't, and swapping it sounds awkward?
By using the we are essentially pinning the description of the noun onto the noun itself. To reiterate the previous point, we're not just talking about any kind of awareness, we're talking specifically about awareness of the things that he did in [the] dark.
What explains why you need to say "in the dark" here?
Here the definite article is working in reverse. When used with some adjectives -- like dark or wild, for example -- the definite article instead transforms it into a noun. While dark is an adjective (With the sun down it's getting dark quickly...), the dark is a noun that describes itself (...and I really don't want to get stuck in the dark). Wild tigers live in the wild, not in captivity. Both the sky and the waters of the seas and oceans are blue (in color) and as such can be referred to as "the blue."
Wait, what about this "idiomatic use" of the that some people have commented on?
Idiomatic use can alter things a bit, but even then they still follow the basic rules. In DrMoishe Pippik's example, he illustrates a difference between British English and American English. The British "to hospital" is describing a general type of place, so it omits any article. One could easily include the article "a" and retain the same meaning (to a hospital).
On the other side of the (proverbial) pond, Americans generally use either "to a hospital", which has the same meaning as its British counterpart, or "to the hospital" which might be confusing if you're not familiar with the American tendency toward linguistic ellipsis. In this case, it is generally understood that the hospital in question to be the most appropriate one; commonly this is the closest one (when seconds can mean the difference between life and death, a longer trip can literally kill you), though it can also assume an associated hospital (a Mayo Clinic Ambulance brings people to Mayo Clinic), or one that's been determined in advance ("The baby is on its way, so we're headed to the hospital [that we elected to have the baby at]").