What do large companies get from a higher market cap?

Typically a higher market cap means a higher stock price. A higher stock price is useful for a few things:

a) Self-defense when the stock price is very low, the entire company could be bought cheaply by a competitor. Some companies will put a high priority on increasing their share price if they are in danger of this happening.

b) Making acquisitions: the reverse is that the company can acquire others and pay with shares. If the shares are trading at a high price this is easier.

c) Raising capital: if the company issues more shares it can bring in a lot of capital. Facebook reportedly sold more shares than they needed to during their IPO, because they realized the price they would get was a great deal for them and they wanted to get as much capital as possible.
- Reputation and influence: a higher market cap typically means that people pay more attention to the company and its executives. Though less tangible, being an industry leader may have many benefits for the company.