I thought they were the same thing except I've just found out they are not!
Using IE8 I can get the same webpage to display in 3 different ways:
Standards mode. The XHTML Strict DOCTYPE is at the top of the webpage.
Same as above except that 'compatibilty mode' is clicked in IE8. Now the webpage is displayed slightly weird.
If I remove the DOCTYPE from the webpage (this forces quirks mode?). Now the webpage is displayed very badly and is not the same as when viewed in 'compatibility mode'. At this point clicking 'compatibilty mode' makes no difference.