Since very little differs in the specification for the two computers, I just installed the drive and booted. It worked for a while, but after a hard reboot (due to other issues) it started to behave strangely, processes were locked up, and I couldn't shut down cleanly. I tried to reinstall Mac OS X v10.8 (Mountain Lion) over the existing installation, but it would not complete. I then erased the disk through Disk Utility and tried a fresh install, without any luck. I'm able to partition it, but the installation locks up.
I then removed the drive and connected it to another Mac with a USB-to-SATA adapter. Over this connection I was able to install OS X. When I tried the same with the MacBook Pro, the whole installation completed successfully, so I reinstalled the drive inside the computer and booted, and it seemed to work until I ran migration assistant, which failed. (The syslog was full of errors of core processes that couldn't read from "(null)".) I then tried to use rsync to transfer my data, but I got a write error followed by "broken pipe". Just to check I removed the drive again, connected it over USB and run rsync again, without issues.
I have yet to try it inside the original MacBook again, but so far my experience is that it works when connected over the USB-to-SATA adapter, but not over the internal SATA connector in the MacBook Pro. The S.M.A.R.T status shows no problem. What could possibly be the reason?