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 :)

Smith's view of specialization is not the same as today's "interchangeable cogs" view of disposable labor. He's talking about skilled crafts- and trades-man who can both use specialized machinery and acquire skill in its use, as well as avoiding the task-switching costs of single-person fabrication. Smith talks of market clearing not in hours or days, but months and years. His view of labor mobility tends to be generational rather than of a person having multiple careers in a single lifetime, as is the present fashion.

And while a given software program on its own won't generally improve performance over time (absent some learning and adaptive algorithms, though these are still in very limited use), in an appropriate (e.g.: Free Software or other iterative / continuous improvement) environment, coders will make incremental changes to code, which is far easier to accomplish when a program does a single, clearly identifiable task (or can be divided into modules which perform similarly).

Smith's use of the "invisible hand" analogy is a grossly misinterpreted and minor element within The Wealth of Nations.

Code that is organized so that it minimizes context switches is ofen faster and more efficient than code that doesn't.

E.g. sometimes this code:

  foreach (x in xs)
      F(x);

   foreach (x in xs)
      G(x);

   foreach (x in xs)
      H(x);

is faster/more efficient than this:

   foreach (x in xs)
   {
        F(x);
        G(x);
        H(x);
   }

I disagree. When I write a program, I find more and more of bugs the more often I run it. Just as the tradesmen refine their skills, I'm refining my program. (Both are our outputs.)

In addition, by being small and focused, my program will expose more of its bugs for each run because a higher proportion of the code is exercised. (Assuming that we're not talking about the exact same inputs, just like a tradesworker has some variability to work within.)