Although the grammar for よう and みたい is slightly different their meaning is the same. You can think of よう as the more formal version of みたい. Both can only be used with verbs.
Let’s look at よう first. The kanji for よう is 様, one of the meanings of which is “appearance”. The structure for connecting nouns is:
|A dream-like world.|
Used in this way, よう is almost always written in hiragana.
|Maintain a family-like relationship.|
To connect the structure with a verb we need to use the target particle.
|Act like a robot.|
|I reckon she’ll be late as always.|
And よう can also be connected directly to the standard form of the verb.
|It seems like he’s absorbed in his studies, eh?|
|It seems she has a lot of friends.|
みたい appends itself to the standard form of verbs, nouns, and adjectives. Grammatically, it’s usage is the same as よう except that to connect it to a noun we do not need the possessive particle.
|Say in a girl’s voice.|
|You don’t look so good. Did something happen?|
|That sounds like a lie, right?|
|You’re like a child.|