I'm going to answer differently from the others because I think it's pretty clear that Smith is different from the other Agents, even in the first movie, so I'm going to answer only with respect to movie one.
First of all, I think Smith himself gives the answer to your question.
I hate this place. This zoo. This prison. This reality, whatever you
want to call it, I can't stand it any longer. It's the smell, if there
is such a thing. I feel saturated by it. I can taste your stink and
every time I do, I fear that I've somehow been infected by it.
Note that everything that makes Smithy unique also makes him more human. He gets angry and emotional, feels desperation, holds deep personal grudges. He desires privacy (hence the moment of surprise when he asks his fellow agents to leave him and Morpheus, and then takes out his earpiece, the physical symbol of his communication channel to them), which shows that he he is aware that his emotional outbursts are somehow wrong.
Conclusion: Smith has spent so much time in the Matrix that humanity has begun to taint him.
Remember, these are programs, but still artificially intelligent programs, with all the capacity for growth and change that this entails. Smith seems to take charge in most situations, so perhaps he is the oldest, and so has had time to be corrupted - to be humanised - by the inhabitants of The Matrix. Perhaps not, and he - as an individual - is simply more capable of change than his counterparts. Either way, the result is an increasingly erratic program that shows signs of that which it despises - humanity.