For me, one of the main differences is
Plot-based campaigns vs Location-based campaigns
A plot-based, linear campaign is primarily based on a story that the party experiences. Usually, there is one main arc, and while there are side-quests, it is generally clear for the players what the main arc is.
As a DM, you will take care that the party is involved in the major plot points of that story, either as actors or as spectators (or they get told about it if they manage to miss it). If they party manages to stray off the railroad, the DM might move scenes or bend they story so the players still experience most of it.
A good examples for this kind of campaign are almost all pathfinder adventures paths.
A open-world campaign is less concerned with an overarching story than with simulating a world; not necessarily mechanically, but the world is expected to go on and stuff happens without the players being privy to it.
As a DM, you might prepare a few locations and NPCs, but since there is no set main story, it's much more easy to improvise; the party doesn't need to do action X and then Y in order for the world to move forward. Its also easier to shuffle things depending on what signals you as a DM get from the party. If they are really interested in a specific NPC or quest, you can easier grow it without worrying about the plot, as long as the world stays (reasonably) consistent.
A good example for this kind of campaign is the LotFP free rpg day adventure Better than any Man.