The answer is exactly on you question. Code is detailed. It need to has every implementation detail, a spec is high level view demoted of all unnecessary details!
It captures the why and what of the system from a business or user perspective. Existing code don't inherently capture the original business problem or design choices that led to that code.
Fair enough! Business and user perspectives are meta-data that explain the intent behind the code. I disagree with the point that intent should be the source of truth, though. Intent, like you explained, is high level and lacks the details to be an unambiguous source of truth.