Cookies and sessions allow storing and retrieving data across multiple requests from the same client in web applications. Cookies are stored on the client side, while sessions are stored on the server side. The key functions for cookies include setcookie() to create cookies, and $_COOKIE to access cookie values, while the key functions for sessions include session_start() to initialize sessions, $_SESSION to access session variables, and session_destroy() to delete session data. Both cookies and sessions associate data with a unique ID, but sessions only last for the user's browsing session while cookies have a longer lifetime.