The difference between At heart and Deep down
At heart is also phrase with the meaning: in spirit.
Deep down is also adverb with the meaning: fundamentally.
check bellow for the other definitions of At heart and Deep down
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Deep down as an adverb (idiomatic):
Fundamentally; in essence; in reality
Examples:
"He seems like an average businessman, but deep down he's an overgrown kid with a necktie."