I thought the monthly subscriptions allowed you to download music without paying for each individual song?

It does. Use the Music app on your iPhone or other iOS device, not the iTunes Store app. See the instructions in Add and download music and video content from the Apple Music catalog - Apple Support

Also note that even when you have an Apple Music subscription it is also possible for you to purchase music by using the iTunes Store app. Rights to listen to Apple Music only continue while you have an active subscription while the rights to purchased music do not expire.