So far, these are the ways I know:

-Cocoa apps' context menu for text always has
Cut, Copy, Paste, Spelling, Font, Speech, Writing Direction, Look Up in Dictionary, etc.

e.g.: In the Finder, being a Carbon app, the Search box does not have a context menu.
(Well, it does, but it's for the Search box toolbar item, not the for the text).
Furthermore, when you rename a file/folder, there is no context menu.

-Cocoa apps' toolbar buttons are click-through; Carbon apps' toolbar buttons are not.

Click-through means being able to click on a button of a window you are not currently focused in.
You're in a Safari window, and you see another Safari window in the back.
Since Safari is Cocoa, its toolbar buttons are click-through,
so you can click the toolbar buttons of the background window even though you're not focused on that window.

(a little aside: Here's an article on click-through.)

Anyone know of other ways to tell the difference?