Korean vocabulary does not match exactly with that of English: there is no one-to-one correspondence between two languages; There is no exact word for 후회 and 미련 in English. "regret" could be a possible choice but it could mean both “후회” and “미련” depending on the context.
'후회' is a feeling of remorse or even guilty for misdeed or mistakes. You did something wrong in the past. You realized that you should have not done that. You wish your misdeed had never happened. If you could go back to the past, you would definitely change the misdeed.
예: 대부분의 흡연자들은 담배를 피우기 시작한 것에 대해 후회한다 (Most smokers regret having started smoke.)
'미련' is a desire for something that you have lost as a result of (usually tough) past decision. But the decision is not necessarily a wrong doing or mistakes; they could be good decisions. Even if you could go back to the past, you were not sure whether you would change your decision; but you miss the result of the decision not taken and feel sad about the result. Let’s suppose a mother have chosen a family over her career. The choice is neither a mistake nor a misdeed. it could be her sacrifice for her children. Given a second chance to choose, she would probably choose the same choice. However, in the back of her mind, she could still want her days in corporate world; she could miss her splendid career; she could feel sad about her lost chance. The emotion of looking back and feeling sad is “미련”.
example: 지나간 일에 미련을 두지말자. (Don’t look back or Don’t regret over the past).
I hope my answer help you understand the difference between "후회" and "미련".