The song limit is really better in Apple, but you are able to add your own music in Spotify.
To listen to it in other devices I had to download it in them.
From what I remember having spotify, the only way to add your own music is to add it to a playlist from your computer, and then sync that playlist locally, so it basically just uses Spotify as a relay for your music on your computer. I'm assuming Apple is like Google Play Music, which lets you add albums and songs to your library that appear the same way as the streaming ones (within your albums/artists/etc) and don't need to be synced locally.
It is inside the app. It is the same process that you do to make playlists available offline. I only did that from my laptop to an android phone, I don't know if it works in all clients.