copula
it seems that; it looks like
Expresses a conjecture or impression based on direct observation or evidence. Polite form of みたい. Often used in conversation.
It seems to be raining.
He looks tired.
copula
Used in similes or metaphorical comparisons, often with まるで. Polite form of みたい.
It's as if it were a dream.
みたいだ is a casual, spoken way to say 'like/similar to', in contrast to the stiff, written-style ごとき.
みたいだ is a casual, everyday way to say 'like/as if', whereas 如し is archaic and appears mainly in proverbs and classical texts.
Derived from the adjective みたい (seeming, like), which itself may be a contraction of 見たよう (looks like). The polite copula だ is added for formal speech.