Thomas,
The Like Box and Like Button are for completely different purposes, though there can be a little overlap, as I'll explain.
The Like Box shows information about a specific Facebook Page. It can be configured to have a Like button for that Page, show recent admin posts to the wall of that Page, and show user pictures who have Like'd the Page. In general, it's a mini-window into a Facebook Page. It's good for promoting your Facebook Page, and that's it.
The Like button is a way to garner Like's for each URL on your site (home page, articles, etc). It has no connection to your Facebook Page, unless you configure the URL for the Like to point to your Facebook Page. That way, you can have a simple like button for your Page without all the extra stuff of the Like Box (stream, avatars, etc).
Both modules work if the user is not logged into Facebook for display. However, if the user wants to click the Like button, if they aren't logged in, a popup will appear for them to login to Facebook.
Some URLs from our site with demos:
Like Box for Joomla / JFBCFan Feature Overview
Like Button for Joomla
Or, check out the
Facebook Plugins information page
which has their descriptions for each.
Hope that explains, but if you have other questions, just let us know!
Alex