Why not make it simple for people and just use one name? How does the "technically different" part affect people?
Because YouTube Red is a brand targeted at people who are using YouTube now and want it to be ad free. Google Play Music All Access is a side benefit for that audience.
(Conversely, Google Play Music All Access is an existing product targeting people who want unlimited music streaming -- while the music use of YouTube is somewhat relevant to that audience, ad free YouTube viewing isn't a core benefit to that audience; so with the YouTube Red coming on board and including, and being included by, Google Play Music All Access, you've got one substantive product with two different brands targeted at different audiences.)
It might, in the long term. The next step after expanding the services offered under Google Play Music subscription and adding a new name for it (without immediately disrupting anything) might well be retiring the old name, moving the whole audio/video streaming product area under the YouTube brand.