It's different in that publicly traded company market caps are usually based on the value of common stock. The valuation touted in headlines for private compani

It's different in that publicly traded company market caps are usually based on the value of common stock. The valuation touted in headlines for private companies is usually the implied value of a new, privileged block of stock. The common stock is not valued at the same amount. But, only IPO all those special classes of stock convert to common stock. So in the success case, yes, valuation and market cap are eventually consistent.