I'm finishing a degree in pure math currently, but I also spent a lot of time working in applied math research projects. Though every discipline draws its own cultural boundaries, the distinction between pure and applied math is often more elusive than we would like to admit. Until relatively recently in the history of mathematics, almost all of math was what we would now call "applied math." (Grant an exception for number theory if you like.) Sometimes the boundaries shift, as well. One of my research interests was motivated by an extremely "applied" problem corresponding to an actual physical system, but grew to encompass central techniques from semigroup and formal language theory, relatively "pure" topics. Remember that even Gauss, the prince of pure, spent hours calculating the orbit of Ceres by hand. (This is supposed to have motivated the discovery of least squares, so it wasn't a total wash.)
It's very difficult to say much more about your situation without specific details about coursework and research opportunities, but it would be fair to say that applied math will give you much more experience in programming. This is not to say that there are not computational problems in "pure math", (there are!), but that these will not be emphasized, and you will have to dig for them on your own. On the other hand, it seems that most folks have an easier time going from pure to applied to vice-versa. There's lots of opportunity for confounding variables here, but that may give you pause.
Ultimately, one of the most useful skills you can cultivate as an undergraduate is the ability to determine answers to the following: "what do I need a gun to my head in order to learn?" If you have interests that span multiple fields and prevent you from exhausting the course offerings in each, that question should motivate a great deal of coursework. For instance, I love automata theory dearly but I never took a course in the theory of computation because I could just read the textbook for pleasure. (Nota bene: this only works if you actually read the textbook). In differential geometry, however, I knew that I would never actually be arsed to deal with Christoffel symbols and the like unless I had a gun to my head in the form of a weekly quiz.
You should learn to recognize your own inclinations and disinclinations, and reroute around them.