I think it's a little bit tricky question.These two words are usually used to indicate a emotion for a person not for a thing. In general, the word "증오해" has a little bit more strong emotion than the word "미워해"  and that emotion has been lasting for a bit long time or will be long-lasting.It means that it's not a momentary emotion for someone.Whereas,we use the word "미워해" when we usually hate someone.This emotion doesn't necessarily have to be long-lasting.It could be temporary or not.It depends on the situation.

Although my english is not good enough to explain it in english,hopefully it helps you.

미워하다 is just dislike...

증오하다 is really really dislike....

증오 is more strong expression

미워해 is generally used

증오해 is high class level

I think 미워해 is dislike and 증오해 is hate

but i'm not sure

미워해 is more like I don't like you. It is often used between a girl and boy.

Situation: a couple fights verbally. (Not serious sitation)

e.g.) Sarah: I don't like you, Paul!

Paul: Why you don't like me? ㅠ_ㅠ

증오해 is more like I hate you!

Tom: I hate a devil. I hate the guy who kills pigs and cows for food.

if you don't like or hate some person or upset about them, you can say "난 그 애를 미워해(or 싫어해).

It means I don't like/hate him/her"

if you will be mad forever and you want to hurt them, you can say "난 그 애를 증오해"