You can also use them to move stone from your mining areas, though I find that use of minecarts to be inefficient. Each minecart holds only five chunks of raw stone, so unless you have enough routes to keep dwarves at the mine site moving stone they'll wander back to where the jobs or the idling areas are and have to walk back to the mine site to reload the minecart. With wheelbarrows, you'll always have dwarves walking there and back again, but they'll always be bringing something back with them. You can also use the number of wheelbarrows to a stockpile to limit how many dwarves are wandering around scooping up lumps of rock, but this doesn't work as well as it could because there's a three-wheelbarrow limit on a particular stockpile. You can get around this by either making numerous identical general stone stockpiles and having them all feed into one main stockpile (via quantum stockpiling) for ease of linkage to workshops or other stockpiles, or by making numerous, more specialised, stockpiles from the start.
The new stuff provided in the hauling update is really more of a method or a means to an end. Asking 'are they essential? Can you go without?' about minecarts and all the other cool stuff is as general as asking those questions about, I dunno, constructions or something.