Well, I'll give just one example of a difference. I cast away all my tatty old clothes. (I threw them away) Don Juan cast aside his lover in favour of a new love. (loved her then rejected her, abandoned her) Actually, they are often interchangeable. I don't see why Don Juan couldn't cast away his lover. However, it doesn't sound right to cast aside your old clothes (bit too emotional/poetic?) Others will supply more examples. I'm off to cast away some old rubbish I have in the garden shed. I don't have any lovers to cast aside, so it's just the rubbish in the shed to sort out today.

Generally cast aside is used as a term for rejecting something, ignoring something or throwing it away. A castaway a person who has been shipwrecked and stranded in an isolated place.