Unlimited gives you unlimited ad-free listening on your desktop/laptop.
Premium gives you unlimited higher qualitify listening on your desktop/laptop, as well as the ability to use Spotify on mobile devices such as (supported) tablets and phones. Premium also includes the ability to download and sync playlists inside the Spotify client so you can listen to them when you don't have an internet connection.
Both subscriptions allow you to use your account abroad for an unlimited period of time as long as you continue to pay for the service (you only get 14 days on free).
Please bear in mind that the Community is not an official Spotify support service. It's a place where we all help each other, whether we work for Spotify or not. So please use your discretion when using the forum.