Page from 2012
Updated Oct 2014
The relationship between the Product ID: on the activation screen and the COA
The License Number that displays on the Windows 7 "System" screen is not the same as the Product Key (although it is labeled as such).
The License Number does appear on the COA (that is the lable that is supposed to be affixed to the computer that the copy of Windows is installed):
W7 Prof License 00371-OEM-9?4?9?3-6?1?4 and there is an option to "Change Product Key"
Last 9 digits on the COA are similar to those in the graphic above but with the 7 is replaced with a 3 (or visa versa) - the product key on the box (15 digit) is completely different.
As the graphics on this page are taken from my own personal copy of Windows 7 Prof that is legally registered and activated with Microsoft I have obfuscated some of the numbers. It is entirely possible seeing as I have now "watermarked" graphics files the Microsoft could trace back to me knowing that I have admitted to installing many copies of this software on many computers.
The screenshot above shows a Zotec motherboard with a "bootleg" copy of Windows 7 Home Premium 32 bit activated with an ASUS product key.