Anecdotal, I've heard no excitement from any of my normal Apple-obsessed friends about using the new service. I find streaming music apps to more sticky than I had anticipated - people have their playlists/history - and they don't want to lose this. My wife was a user of MOG (one of the few), then of Beats post-purchase. She hates the Beats interface/app, but she will not change to another service. Spotify has another important sticky factor - the desktop app - which allows you to play back both what is on your hard drive with the streaming music.
This is not to say Spotify isn't doomed long term - they have, in my opinion, a fairly unsustainable business model that continues to exist only because of VC funding. They need to create a more multi-stream business - just working with the labels will put them into the ground. My guess is that they continue to evolve into a more complete media platform, and that they sign artists directly to get to a real margin.
But to say that Apple will crush them reminds me of the whole Netflix/Blockbuster comparison. Now Blockbuster was no Apple - but everyone thought that they, as the incumbent, would crush Netflix. That didn't happen - because Netflix was there first and Blockbuster launched a very me-too competitor. Spotify is also focused on winning this - it's all they have - whereas Apple Music is a tiny part of Apple as a whole.
Now, this could also play out like Google Maps - I believe Apple Maps now crushes GM in terms of usage - so Spotify will likely have trouble going forward if Apple Music is installed by default on every handset.