I've never read The Wealth of Nations, but Smith wrote well before the industrial revolution. So when he spoke of workers, he was referring to tradesmen and craftsmen. For them, specialization does allow them to get better by performing the same tasks over and over again.

In a way, the industrial revolution just changed the scope of his arguments, but I think the basic idea still holds. Now, instead of thinking in terms of a single worker, we could instead think in terms of a single factory. Each factory, comprised of many workers, would then be able to perform one specialized task well, and trade for the rest. It would be up to each factory to organize itself in an optimal manner.

In thinking in terms of a program, it wouldn't be the program that gets better, but the programmer. If they write a program to perform task X multiple times, that program will inevitably get better with each iteration. So, ideally, what you end up with is a series of optimized programs that can work together in different configurations in order to perform different tasks in the most efficient way.

At least, that would be the theory. I'll leave discussions of the invisible hand of the market for later :)