How do I show a different homepage to logged in and non-logged in users?

If you're comfortable adding functions, this does the trick.

    /**
     * If a user is logged in, tell WordPress to use 'page' on front page of the site
     * @param string $value
     * @return string
     */
    function fn_set_page_as_front_for_loggedin_user( $value ) {
        if ( is_user_logged_in() ) {
            $value = 'page';
            //page is set as front page
        }
        return $value;
    }
    add_filter( 'pre_option_show_on_front', 'fn_set_page_as_front_for_loggedin_user' );

    /**
     * If user is not logged in, set our static page to act as home page
     * @param $value
     * @return int
     */
    function fn_set_context_based_page_on_front( $value ) {

        if( ! is_user_logged_in() ) {
            return $value;
        }

        //for logged in user, use page id - in this case 56
        return 56;
        //change with your own page id.
    }
  add_filter( 'pre_option_page_on_front', 'fn_set_context_based_page_on_front' );

Rarst's solution is indeed the simplest way to do it and the Members plugin also has heaps of other advantages.

You can also do it without a plugin, by using the is_user_logged_in() function

More info on http://codex.wordpress.org/Function_Reference/is_user_logged_in and http://codex.wordpress.org/Function_Reference/get_currentuserinfo#Parameters

Home page only or whole site?.. I assume latter as it makes more sense.

The simplest way I know is to use Members plugin and enable Private Blog option it has.