"You know differently" is perhaps more grammatically correct but many people say "you know different", using an adjective instead of an adverb. Compare "you should know better".
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Anon nailed it. To know "differently" means the mechanism of your ability to know differs from other people's.
Like "thinks differently". Adverbs adjudicate how and how well you perform the action indicated by the verb.
If you want to say you understand that someone's wrong, you'd use the adjective - "you know different"