If you really want me to explain, I have to start with the difference between 은/는 and 이/가
It's a hot summer here and a motor in my laptop is burning so let's make it quick and simple, shall we?
Whenever you see X은 or X는 in a Korean text, you can say to yourself, "Oh boy, this guy is about to explain(or talk) something about X".
In the other hand, X가 or X이 merely dictates that X is a main agent(subject) of the sentence.
사람은 밥을 먹습니다. (사람, let me explain about 'people', they eat.) (a little assertive/subjective/explanatory sense)
사람이 밥을 먹습니다. (people eat. simple fact!) (a little dry/objective/factual sense)
Alas, how am I supposed to teach this subtlety of those two differences! Well, I just overgeneralized the hell out of it, so don't buy it too much. Anyway back to your question:
1. 좋아하지 않아요. (I don't like it)
2. 좋아하지는 않아요. (The thing that I like it, well let me explain, it is not so true)
Simply put, #1 sounds like you hate it. Of course, [-]LIKE implies [+]HATE. (Maybe hate is a too strong word. But you see what I mean.)
But #2 somewhat suggests that it is true that you don't like it, but that's all. You can translate: "It's not that I like it... (but it's okay...)"
1. 방이 깨끗하네요.
2. 방이 깨끗은 하네요.
#1 The room is clean.
#2 The thing about the room being clean, well, let me explain, it's true... (but it's too small, or, but it's too expensive, or, but there is a dead hooker, etc)
1. 너 오늘 참 예쁘다.
2. 너 오늘은 참 예쁘다.
#1 Today, you look so good. Girls love to hear that. Use it.
#2 Today, (big pause) well, let's talk about today, not any other days, you look good. (Which strongly implies she doesn't look good other days, but today is bearable)
To me, It is hard to say the difference between them.
They have almost the same meaning.
But I'd like to say '좋아하지는 않아요.' is more flexible than '좋아하지 않아요.'
If someone ask you 'Do you like Orange?',
In this situation, Let's pretend that you don't like Orange!
You can say 좋아하지 않아요 or 좋아하지는 않아요.
First, If you say '좋아하지 않아요', NORMALLY, The conversation will end.
But, If you say '좋아하지는 않아요‘, Sometimes the asker will say 'So, How about Kiwi?', 'What kind of fruit do you like?' and others.
Because '는‘ has some tiny aftereffects(I couldn't fine a proper word, But, In this writing, aftereffects means '여운’in Korean)
And You can add your opinion as well after saying '좋아하지는 않아요' like the following sentence.
(오렌지를 좋아하지는 않아요, 그러나 싫어하지도 않아요 I don't like Orange, But don't dislike Orange, either)
** A certain main will ask more, even though you say '좋아하지 않아요.'
** I am telling you about the feeling that Those sentences have respectively
It is just my opinoin, Let's wait for another person's opinion!