To date, playing full songs on Apple Music has meant firing up iTunes or the mobile app. You could listen to samples on the web, but that wasn't much different than playing iTunes Store clips years ago. This appears to be changing, though. Reddit users have discovered that Apple Music's embeddable web player now plays complete songs as long as you sign in to your account. You can even add albums and playlists to your library without having to leave your browser.
This isn't a full-fledged web player à la Spotify. You can't browse the catalog, create playlists or see what your friends are playing. There's no guarantee you'll see a dedicated web client in the future. However, it's still miles above the previous functionality, and suggests that Apple is taking web playback seriously where it previously served as a marketing tool for the company's native music apps. Don't be surprised if Apple talks about this at WWDC.
In some ways, Apple might not have had much choice. If it's going to catch up to Spotify, that means offering players everywhere Spotify is an option -- and that includes the web. You may be more likely to subscribe if your favorite review site can embed Apple Music's version of an album instead of alternative services. This also helps Apple offer some form of its service on devices where installing an app just isn't possible, such as a locked-down work PC or a Chromebook.