(Not a teacher)
Woeful means something is full of woe, or unhappy. A woeful person is an unhappy person. A woeful melody is an unhappy melody; one that evokes sadness.
Awful most commonly is used to mean unpleasant, really bad. An awful person is a mean person. An aweful melody is a melody that isn't pleasant to listen to.
Awful can also be used to mean 'full of awe', or creating awe in oneself - "The awful image of the Earth as seen from the moon." for example. This is much rarer than the first meaning, although it is the literal meaning of the word.
Informally, awful is also used as an adverb to mean 'very' - 'He was an awful good artist'.