I'm not sure they're very different, if at all. I guess a public valuation is a bit under what you'd actually need to pay to acquire it outright.

Most simply, value is a non-fixed inherent property that can change depending on the asset / liability and the entities involved. A valuation is an actual estimated value given some conditions.

Simple example: Celery has no value to me, because I find its taste disgusting. However, I happily pay for bundles of cilantro, which in turn have no value to others that find its taste disgusting.

In the case of a publicly traded company, the conditions are "if all shares were immediately liquidated at the current price". In the case of a company buying that publicly traded company, an actual value can be arrived at, due to having a fixed set of assets, liabilities, and participating entities. And, as you pointed out, those two numbers rarely match up.

Well, I don't think it's entirely uncommon for a complete set to command a premium.