Specifically, the Riemann integral of a function f: [a,b] → R exists iff f is bounded and the set of points x at which f is discontinuous has measure 0.
The chief advantages of the Riemann integral are its extensions to the various improper integrals. The limit-based formulation is a particularly nice fit, as it uses the order-based nature of the Riemann integral. The Lebesgue integral, to give the most famous and commonly-used alternative, depends on a measure -- and loses many of the benefits of the complete total order we have on R.
Doing integrals in (infinite-dimensional) Banach spaces is a lot more difficult. Given a "box" of side h, it is not even possible to define a "volume" of the box (the volume of such a box of dimension d is hd for finite dimensions, so we'd get either or infinity for infinite dimension -- and neither will do).
You really need a measure to integrate, and then you might as well do a Lebesgue integral. Even the "box volume" problem above is really a problem defining a measure consistent with the norm.