Finally found what the hell is the difference between Live and News Feed. The horse’s mouth says:

News Feed - When you log into Facebook, you’ll see the most interesting things that happened in the last day in the “News Feed” view. News Feed picks stories that we think you’ll enjoy based on a variety of factors including how many friends have liked and commented on it and how likely you are to interact with that story.

Live Feed - Once you’ve caught up on what you missed, you can click through to “Live Feed” to see what’s happening right now. As long as you remain logged into Facebook, you’ll continue to see posts and activity from your friends in real-time. You can edit what appears in this view by clicking “Edit Options” at the bottom of the home page.

Translation:

News Feed: some random stuff FaceBook thinks is important to you (how likely you are to interact with that story?! - how does that get calculated?!);

Live Feed: EVERYTHING.