Screen -- Your screen: size value changes with your monitor size.
screen.availWidth //There is no screen.height
Window or Document -- The Browser's window (the browser viewport, NOT including toolbars and scrollbars). Ignores the invisible scrollable part if the page is scrollable. Use 'window' for IE9 and above, its simple.
window.innerHeight //IE9, Chrome, Safari, Firefox
document.documentElement(or body).clientHeight //IE 8,7,6,5
Note: Window and Document are not the same. Document object (from DOM) is a property of the Window object (from BOM). But give out the same size.
From W3Schools, I would like to think that 'Window' is the notation for the new browser versions (IE9, Chrome, Firefox etc) and 'document' is for IE 8,7,6,5.
http://www.w3schools.com/js/js_window.asp, and also tested the above with a simple aspx page on differently sized monitors.