In the short-term, micro context, like most exclusive Apple apps, it's one more reason to buy Apple devices and lock existing users into upgrading to Apple devices. Switching apps is a hassle, especially if there's no easy ways to migrate data.
In the broader context and rough terms, it's against Apple's interests that web apps in general take off over native apps. In theory, there's no reason web apps shouldn't be as capable as native apps for 90% of workflows (playing music being well in that 90%).
If web apps, at large, take off:
1. The App Store loses relevance, and so Apple loses influence over developers, distribution, and 30% piece of the cake.
2. Apple loses platform lock-in, since web apps run everywhere. It'd be a no-brainer for companies to have one team developing for the web, instead of one for the web and one or two for mobile.
I don't doubt if Apple wanted they could form a team with enough expertise to build solid web apps and make their services gain marketshare, but the experience would either have to be subpar vs native (like iCloud.com is IMO) or solid enough that they would unwittingly prove web apps are equally capable to native apps, and so create demand for more web apps, more progress in that space, and stray away from an advantageous position for them.
Oh, go figure. I stand corrected on my first point about immediate platform lock-in. Longer-term, I still believe Apple is refraining of heavily supporting the web to keep overall control of its platforms.
With the API integration it seems pretty straight-forward to make one, so there's got to be a strategic reason they don't want to?
Maybe they want to encourage people to use native apps wherever possible, since they have more advantage over Chromebooks there?