What can and can't a private bank do with their cash deposits?

I am a little curious about this. Private banks usually loan out their cash deposits to customers at a higher interest rate than they pay out for those deposits. Can a private bank invest this cash deposits in the stock market or any other type of investment? Also do the same rules apply to publicly traded banks?

I don't know for sure. But keep in mind that banks are highly regulated even if they are not publicly traded. In the US, the main regulators are the FDIC, the Federal Reserve Bank, and the state banking department of whatever state the bank is chartered in. The regulators determine to a large degree what types of financial instruments banks can invest in.

Most US banks invest in bonds and some derivatives like swaps. As far as I know, investment in the stock market is still prohibited (but I would verify this before reporting it as fact).

Either a private or public bank can invest their cash deposits in another kind of investments. Private banks usually loan out to customers at higher interest rate for those customers who have a lower credit rating where public bank will reject their loan. For this a higher interest rate is charged upon them. Hope it helps to clear your question.

1) Yes.
2) No.