If you play any music on your mobile it will stream as in the past unless you decide to make it available offline. You can now make music available offline from playlists as before, Artists and Albums.
I posted the message below to another discussion. Someone asked me to add it here, as it is more relevant to this discussion than it was to the other one. Although it looks like most people here already figured these things out, I figured I would still post it:
"Actually, you can see which songs you've saved when you're on an artists page. Just hover over a song and if a little check mark pops up where the length of the song was (far right), then it means you saved the song to your library of songs. If there is a plus sign, then it means you haven't saved it and if you click; clicking on the plus sign will save the song. If there is an "X", clicking on that will remove the song from your library. Kind of a pain because the stars were obviously much easier to see, but doing it this way doesn't take that long. The only way I've been able to figure out how to star a song is to right click, scroll down to save as...and click Starred. Although I don't really see the point in the Starred option now that they've gone and changed everything. Spotify is getting lame."
They haven't enabled offline mode for Your Music on desktop yet but it is coming. On mobile, it's available from Artist - will download all tracks for the particular artist and for Albums. I think the dev also said that we would be able to downlod from Songs soon. I know this is a bit rough around the edges but, hopefully, YM will replace playlists totally apart from mix playlists once they've rolled out these changes and enabled us to add local files to YM.