TYB
The first thing we know about The One is that it will be a person born inside the Matrix. In the strictest sense, Thomas Anderson isn’t born in the Matrix, he’s born in a gelatin-filled pod so his body can be harvested for energy, and his brain is plugged into the Matrix. Agent Smith, on the other hand, as part of the Matrix code was actually created (or born) inside of it. The One is also able to manipulate the code of The Matrix. While Neo can do this, so can Smith. After Neo "unplugs" Agent Smith from the Matrix at the end of the first film he’s slowly able to change parts of it himself, including making multiple copies of himself to fight Neo.
Eventually, he ends up pretty much taking over the entire thing.
In the end, Neo and Agent Smith fuse together, which means that Smith joins with the source, which is the stated eventual goal of The One. So basically, Neo isn’t so much The One, as he is the conduit by which Agent Smith is used to be the One. It pretty much happens against Smith's will, though willingness is not a prerequisite.