Apple's biggest advantage isn't that Apple is "Apple"; their biggest advantage is that if you own an Apple device, you already have Apple Music. You can't uninstall it or run away from it.
Using every other service requires downloading that service's app and while that isn't a huge barrier for those of us in tech, that is a HUGE barrier for everybody else. Plus think about all the grandparents, soccer moms, and teenage girls that use iPhones and don't really consider any other phone. Those people probably aren't the people that are using Google Play or Spotify, but now they have Apple Music on their phone and ready to go. With all the attention Apple Music has been getting from that whole Taylor Swift Fiasco, there's no doubt that almost everyone has heard of it already and eagerly awaiting its arrival.
If you're a techie and you own an Apple device, then why partition your music selection with Spotify? Apple Music has almost every song Spotify does plus Apple has a better relationship with the record companies, so you know you're money isn't going to a company that's under some serious pressure to change their business model. Plus Apple Music has music videos and a social component for artists to reach out to their fans - which is kinda slick in my opinion. If none of that moves you then okay, try it for free for 3 months. Three months!! No matter what you think about Apple Music's competitors, why not stop paying them for a month and try out Apple Music? I'd wager many people aren't going to move after that.
Apple Music's future success won't depend on the service it's self, but on fitting into Apple's ecosystem and Apple's ability to reach the general public in ways that other streaming services can't. Plus, iPhones are the best selling phones in the United States so a lot of people will have this service already. I wouldn't be surprised if by the end of the week, more people have Apple Music on their phones than Spotify.
Spotify is in serious trouble. That's BEFORE we start discussing how record labels want Spotify to kill its free tier and how many users will flee the service when that happens.