The real competition isn't between Mac OS and Chrome OS, it is between Windows apps and Web apps. Both Google and Apple have a common interest in enabling rich web apps that can displace Windows apps. For Apple, the goal is to reduce the application lock-in that ties users to Windows, allowing Mac OS X, iPhones, etc., to be the client instead of Windows PCs (and Apple wins). For Google, the goal is to drive everything to the web so that Google can be the server; if people use Gmail, Google Docs, Google Calendar instead of MS Office and Exchange, Google wins.
You need to think of things from the perspective of the companies that own the music. They're far more willing to participate in an "experiment" on a test market (the Mac user base) than on the entire PC market. This means that Jobs could get them to agree to things for Apple Music Store that they haven't ever before been willing to agree to. And now that the AMS is perceived as a massive success, Apple's positioned perfectly to extend the store to Windows.
As a second issue, Apple's in the business of building a platform and a brand. As long as Apple can ship category winning products for the Mac first, get massive publicity, then ship a version for Windows that's forced to be inferior due to the limitations of Windows, that wins on all fronts. Apple's product makes money, the Macintosh platform looks better, Apple gets great PR, Apple gets tons of sales outside of the Mac market, and ultimately Apple converts more customers to its platfom. And while Apple loves to sell iPods and music to non-Mac customers, they love even more to convert customers to the Mac because they turn into loyal, life-long customers.
Also, the music industry runs on Mac's, not PC's. And they like and respect Apple.
Finally, don't underestimate how much the music industry won't allow MS to gain any control over their business. They're not stupid people, and they've seen what MS has done to everyone else in its value chain (i.e. buy or kill them). The last thing they're going to do is give MS a foothold in the music business.
Counterpoint: Chrome OS Is Exactly Where Apple's Headed
Apple Should Have Released The Music Store For Windows First