みたい is so-called 話し言葉(spoken word)、while よう is 書き言葉(written word). Though it doesn’t mean that よう is not used in speech.
The main difference(next to formality) is the grammar construction.
There has to be の if the noun precedes よう, or な if it is an な adjective. みたい doesn’t need that.
For example, construction ようにみえる can be used with みたい without changing meaning.
Though you have to remember there are other よう uses which cannot be replaced by みたい. Like ようになる、ようにする、よう(so that/in order to),ように言う・頼む・命じる。
@Putyouinmyoven
No worries! 
I will update those lessons soon, so the answers with the closest similar grammar points would be accepted, though still ask for a correct input.
Also, learning the nuance is very important, if it is not done in the beginning(especially since differences are easy to remember) it might cause a student to pick up bad habits.
Like @seanblue said, some have pretty strict rules, like demanding being direct observer.
