One reason is catering to the potential of the display format.
Looking at YouTube specifically:
- On the desktop, there is room to show many more videos, in a two-dimensional format. This allows the videos on your home page to be organized into categories like "Watch it again", "recommended", and specific channels that may interest you.
- On a smartphone, there is just a single list of videos, and only a couple of videos appear on the screen at a time.
It seems logical that the content displayed will be somewhat different, to maximize benefit to the user in each context.
That said, my recommendations on both are pretty similar. It does seem like a UX failure if the content shown on one is radically different (assuming you are signed in with the same account, of course).