One way they could know is when the OS talks to the mother ship when it downloads an update, for example. Assuming the OS copy is registered they know where to find you. Even if you didn't register, or used a bogus name/address, they'll have your IP address and can track you down the same way RIAA does.
The reason Apple isn't likely to go after individuals is that the benefit they would get isn't worth the cost. You can't collect boatloads of damages for infringement from an individual who is likely judgment-proof (i.e., not wealthy). That said, the breach of the EULA still renders the installation and/or use of the operating system an act of copyright infringement.
The reason Apple cares about running their OS on other hardware than theirs is: (1) they didn't make a profit on selling the hardware and (2) they lose control of the user experience. OSX running on hardware other than what Apple uses might have issues that would lead the user to believe that OSX is a POS and they tell their friends and before you know it everybody thinks OSX sucks.
Incidentally, part of the reason that Macs are more stable than PCs is not that Apple's programmers are better than Microsoft's, but rather that Apple has compete control of the hardware and software and can extensively test the relatively few combinations, whereas Microsoft has no control over the hardware or even many of the drivers and can't possibly test even a statistically significant subset of the combinations.
Apple has sued commercial entities for selling PCs with purchased copies (notice I didn't say legal copies) of OS X pre-installed. IIRC, Apple won but the result is on appeal, where, in my opinion, they will still win. It may have been finally resolved and I just missed it, though.
