The Desert of the Real (a.k.a. Where the Sun don't Shine)
If we want to get technical about it, this is the only physical setting of The Matrix. Everything else that happens takes place within the minds of the characters. In fact, the entirety of the movie is confined within the Nebuchadnezzar, except when Neo is briefly in a power plant before he is rescued.
While this may be true, there's no getting around the fact that stuff goes down within the Matrix itself. So let's talk about some of the main differences between the Matrix and the Desert of the Real, starting with everyone's favorite hovercraft.
The Nebuchadnezzar is pretty grimy and beat up. It really fits in with the rest of the "real" world aesthetic they have going on: the whole ruinous city and blackened sky thing. The blue lightning really brings it all together.
If we go inside the Nebuchadnezzar we see flickering lights and worn down chairs. The living quarters appear submarine-esque; they seem hard and tight and not particularly comfortable. There are also a lot of wires running around out in the open and the main console with all the computers seems a bit slipshod.
Even though we're about two hundred years in the future, this hovercraft is nothing like the pristinely clean spaceships of other science fiction works, like Star Trek. Of course, they're driving around their ship in the service tunnels and sewers of ancient cities and we're pretty sure they haven't installed any showers. Smith might think the Matrix stinks but he should be glad he's not aboard the Nebuchadnezzar.
Mega City (a.k.a. The Set Down Under)
If you really think about the Matrix as a setting, there is really only one part of the Matrix that we every see: the metropolis known as Mega City. It's actually unclear whether there are other parts of the Matrix.
This city is portrayed as a very modern, corporate kind of world. It's full of skyscrapers and big, busy streets with lots of people. What's interesting, though, is how clean it appears. Aside from Room 101 where Neo lives, most other scenes have a very minimalist, orderly feel: the office and cubicle Neo works in, the building that Morpheus is taken to, the building where the fight scene occurs with its pretty columns, and even the rooftops. Everything appears neat and tidy… which seems to be the whole point.
There is one play that sticks out like a sore thumb, however: the old hotel where Neo is introduced to Morpheus and where Cypher betrays the group after Neo goes to the Oracle. This building is an intermediary between the two worlds. It's a mix of real world dirty and Matrix clean. It has a classy antique feel—dig that checkered staircase and the large armchairs—and, while it may be dusty it's certainly in better shape than the Nebuchadnezzar.
Seeing as this building acts as a kind of portal which Morph and the crew use to enter and exit the Matrix, its aesthetic makes sense.