Private banking (PB) encompasses an array of services for high net-worth individuals. That might mean personal banking, but might also entail sophisticated services: currency hedging, asset disposition, tax advice, portfolio management, and mergers and acquisitions on a smaller scale.
Investment banking/corporate banking (IB) provides services, advice and products to large corporations around the world. That, too, might mean basic corporate banking, but might entail complex debt or equity financings, takeover defense, project finance, convertible-bond offerings, or equity share repurchases.
But sometimes the two intersect. The owner of a thriving, growing private company decides to go public. Private bankers and investment bankers will be involved–to advise the company on its IPO and to advise the owner and other shareholders how to manage its ownership of new public shares or new found wealth.