They both come from 어떠하다 (adjective for 'how / in what way' in a question, and 'in a certain way' in a statement), but 어때 is a sentential form while 어떻게 is just an adverb.
- 어떠하다 + -아/어/여 (declarative or interrogative sentence ending) -> 어떠하여 -> 어떠해 -> 어때.
- 어떠하다 + -게 (adverbial suffix) -> 어떠하게 -> 어떻게 (adverb for 'how', 'in a certain unnamed way).
So 어때? can be a one word predicate (in Korean adjectives work much like verbs), as in 이 영화 어때? (How is this movie?), 맛이 어때? (How does it taste?), 그 여자 어때? (How is that lady / How do you like her?)
Since 어떻게 is nothing more than an adverb, you need a verb to phrase the same idea, so 이 영화 어떻게 생각해? (What do you think of the movie?), 그 여자(를) 어떻게 생각해? (What do you think of her). 어떻게 생각해? is the most common way of asking someone's opinion about something, similar to 'What do you think of something?' (English stresses 'what' (the idea) while Korean uses 'how' (the way you view it) to phrase the same idea. 어떻게 생각해? and 어때? may be considered loosed synonyms.
어때 is not so specific, so its use is limited to those 'How is it?', "How do you like it?'-type of sentences. To ask about more specific things, you'll need 어떻게 with a verb, as in '서울 아트센터에 가려면 어떻게 해야 돼요?', 어떻게 가든 부산까지 두 시간 이상 걸린다 (No matter how you do it, it takes over two hours to go to Busan).
Here's some more examples.
1 이 영화(는) 어떄? = A) How is this movie? (Asking someone who already saw it)
B) How about this movie? (Suggesting to see it)
2 이 영화(는/를/에 대해) 어떻게 생각해? = What do you think of this movie? (correct, but might sound too serious - could be appropriate when reviewing movies)
1 어제 만난 사람(은) 어떘어? = How was the person you met yesterday?
2 어제 만난 사람(은/을/에 대해) 어떻게 생각해? = What do you think of the person you met yesterday?
(#1 is asking more loosely and broadly (how did they feel in general about the person. #2 is more specific about their assessment and opinion)
1 거긴 살기가 어때? = How is life where you're at?
2 (This kind of sentence is not as natural with a specific verb)
(어떻게 used with a different verb)
1 이 경기(를/는) 어떻게 보셨습니까? ("How did you view this game / What is your take on this game - as a newscaster might ask a commentator)
2 이 경기(는) 어떠셨습니까? ("How was this game (to you)?" - more informal)
(어떄 can also be used as a retort, like "So what?" or "What does it matter if ...?")
1 빨간색으로 입으면 어떄? = What does it matter if I wear red? (= it is fine) - here the ending is not raised.
2 (This can't be phrased with 어떻게. You can say 빨간색으로 입어도 괜찮아 = It is okay to wear red)