The code imagery, I think, is largely symbolic, but Neo really does have a unique perspective on the Matrix.
As Morpheus points out early on, even knowing that the Matrix is not real doesn't help the redpills beat the Matrix' programs, like Agents. The Agents have the benefit of being completely disconnected from the pseudo-reality of the Matrix, and thus aren't constrained at all by the "rules" of the world. The humans, even though they know its not real, can only progress so far into convincing their own minds.
Neo, at the end of the first movie, breaks past that barrier, and is finally able to fully grasp the illusion of Matrix while jacked in. This allows him to do things that no human has been able to do - defeat Agents, fly, etc. He sees the world he's immersed in the same way that a disconnected human sees the world on the monitors, and has the same control over his environment that the machines do - that of an outside entity looking in.
Of course, @PeterParker's answer is also part of it. By seeing the Matrix code Neo can more easily understand how the virtual world is being formed, and alter it. He can see, and rewrite, in real time, the flow of information that represents the virtual world around him. I think the debugger analogy is an excellent one.
But I think the choice to show the same falling-symbols visual as that on the external monitors, is meant to show Neo fully breaking free of the boundaries of the Matrix in a way that no other connected human can.
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