An analogy:
If you record a programme off the telly (onto VHS, DVD, or a PVR) you're only acting legally if you do so for 'time shift' purposes only. You're allowed to watch your recording once but then you must erase or destroy the recording. So I', breaking the law because I've got some VHS tapes of James Bond films, recorded off-air about 20 years ago. Similarly if (say) you recorded the the last series of Strictly Come Dancing (and you've watched those recordings) you're acting illegally if you've kept them.
You're allowed to watch Youtube videos as they're 'streamed' but not, officially, to keep them on your hard drive. However most people see it as akin to keeping old telly programmes on tape or disk (particularly if those recordings aren't available for purchase anywhere, so nobody is losing any money).