I would imagine it’s pretty similar to most corporate boards.
The board members represent the shareholders and collectively govern the company’s big picture decisions through majority voting. That could include major strategic decisions like investments or acquisitions or selling off parts of the company - anything with a lot of risk. Also hiring/firing the CEO.
The chairman of the board typically has no more authority than the other board members other than being accepted as the one to lead and organize the other directors. They all have one vote. The chairman usually is the primary intermediary between the CEO and the rest of the board.
Tim will probably work closely with the new CEO and will probably serve as a mentor. I don’t think Apple’s culture would allow for an experienced former CEO from another company to run the company. They probably want someone internal who knows the brand intimately, who respects the corporate identity and who will grow into the role.