Does Apple Music let you hear the whole song on iPhone and iPad? (instead of just 30 seconds or 1:30 of sampling)
I've never had a song not play through on iOS or OS X via Apple Music. My guess is network congestion is the culprit when you don't get a full play. 

Another possibility is your iTunes account is in a geographical region where Apple Music doesnt have broad coverage of the songs in Apple's total catalog. In at case, those songs that are for sale somewhere in the world, but not in your region, might have previews instead of full plays. 

For me in the us, Apple Music gives me all the music I care to consume for the subscription period across all decides that are signed your Apple ID. The notorious artists like Prince simply don't show up so I haven't found a song that' soft sale but not for streaming.