Tidal, Deezer

Could someone with experience with music subscription services please advise. My confusion is that these music services advertise that you can download music for offline listening. Wouldn't that be putting music files in your storage...creating a music library? I have had itunes and rhapsody, both of which you can download music files into your computer. Surely there must be others you can do the same? Thanks for any advice
I have Spotify and Tidal "premium" subscriptions.  Haven't done the "download for off line listening" with Tidal, though I think it is very similar to that offered by Spotify.  With Spotify, YES, you can download whole albums (or individual tracks) to your device for off-line listening.  This allows listening to saved music while out of range of WiFi or not connected to cellular.  With Spotify (not sure about the Tidal requirement) it is necessary to log on to Spotify periodically (at least once in 30 days, I believe) in order to keep the downloaded files.  Of course, their app must be on the appropriate device (e.g., iPhone, iPod) to store and access these files.  I'd also note, you don't "own" the files in any way that allows you to copy them or export them to other storage media (short of using pirating software).  I haven't bothered with downloading in Tidal since I mainly use Tidal when connected and "streaming" through the main two channel system where the improved sound quality is evident.  Tidal SQ advantage is just not evident outdoors while listening with earbuds.  In either case - both Tidal and Spotify - I have saved dozens of albums to personal music libraries that are accessible every time I log on to the respective service BUT not all of those saved albums have been downloaded for off-line listening.  Hope that is helpful.
Keep in mind that the download option is only available for mobile devices and not PC or Mac.
Thanks everybody. So, if you "download" into a library for offline listening from tidal or spotify, if and when...let's say...Brian Eno is removed from the streaming list for licensing disagreement, is it also gone/removed from that library? If so that's no good!

Also, again, with itunes or rhapsody, you can download music into your computer. Why they allow this I don't know, but are they the only ones that do that? I mean, you can put the actual music files physically into your computer and have your own library...and do it for a $9.99 a month subscription. Trouble is, it is in alac (not good for metadata right?) and lossy.
Correct. You aren't getting a high fidelity output file.

Though, I have to say having the ability to cue up just about any album I want makes streaming so delightful. 
If you really want to have the music saved to your catalog, then just buy it. -Like we used to do.