What's the difference between "stared at" and "stared off at?"

What's the difference between these two sentences?

Mary stared at the distance

Mary stared off at/into the distance

Solution 1:

Per comments from Carlo and Janus, normally we only stare, look, peer, glare, etc. at things we can actually see.

We don't normally think of seeing things like distance, darkness, gloom, murk, etc. These are really words for the medium through which we [might possibly] see something.

There's scope for a degree of uncertainty here though. Although we normally stare at the horizon (which can be "seen"), and stare into the darkness or stare into the distance (abstract things that can't be "seen" as such), in all cases the links show a few instances using the "wrong" preposition.

In some cases (He looked at/into her eyes; She stared at/into the flames), both prepositions are possible (using into implies trying to discern something within or beyond the eyes/flames).

There's little difference between stare into the distance and stare off into the distance. Arguably, including off (less commonly, out, away) emphasises that you're not looking at anything more relevant that might be nearby (such as a person who's talking to you). But it's a fine distinction.