However, in the public cash equity markets there is an actual benefit to having a "large" market cap.
note: Here I'll use market cap for public companies and valuation for private companies as measuring roughly the same thing even though they have some differences.
One of the greatest tail winds a stock can have is to be in a popular index, this means that ETF's will be a forced buyer of their stock.
Most indexes have a market cap component and the amount of money tracking large cap indexs dwarfs the amount in small cap indexs so a larger valuation is obviously useful.
It can also help public companies borrow as debt to equity is a ratio that most people consider when choosing to loan( buy bonds) from a company so again market cap matters.
It also helps a public company pay its employee's in restricted shares as these can be cashed out immediately by employees for actual money as the shares come off restriction so again a growing market cap has the benefit of giving employee's raises without any money leaving the company.
Microsoft used this better than anyone during the 80's and 90's.
It's less clear to me why a valuation of over $1B matters to a private company though.
Even stock options don't really seem to matter as a reason for wanting a larger valuation as typically employee's can't immediately convert options to cash. In that case only the IPO price(hence public valuation would matter).
Is there any reason other than signalling that this company belongs with the likes of Uber and Airbnb?
EDIT agree with the child posters but they don't really answer the question of why $1B is important for the valuation in a way that, say, $990M would not be.