Why does the weak nuclear interaction have a shorter range than strong nuclear interaction?

They are mixing two different things here.

The strong force does not work between protons and neutrons, it works between quarks. As a side-effect of the way it works, it also constantly creates new particles, mesons. This particle-creation process is conservative in that if you consider all of the particles that are created, their momentum, charge, spin and so on all add up to zero.

It's all those mesons interacting with each other and the quarks that give rise to a second force, the nuclear force. It is the nuclear force that "acts on protons and neutrons to keep them bound to each other inside nuclei", not the strong force. Of course, one is the result of the other, so half full sort of thing... and that's why you see it called the strong nuclear force, or the strong interaction or all sorts of other names just to confuse things.

The distance that the nuclear force operates over is simply a function of the mass of the mesons and the uncertainty principle; all virtual particles with mass have a maximum lifetime, and if you simply see how far these mesons can go in that time, presto, you get a distance.

UPDATE: I realized I missed the closure.

The weak force is also mediated by massive particles, the W's and Z's. Like the mesons in the nuclear force, they are thus subject to the same range limitations due to the uncertainty principle. However, the mass of a simple meson like a pion is about 100 MeV, while the Z is a whopping 90 GeV. That's heavier than an entire iron nucleus! Now it might sound odd that such a heavy object can be created ex nihlo inside something like a helium nucleus, which is way lighter, but that's the whole idea of the uncertainty principle, for a very short time this is allowed, and that's why it's range is so short and the reaction is so rare in comparison.

The range of the weak interaction is directly related to the mass of the gauge bosons, which they acquire due to the (electroweak) spontaneous symmetry breaking that gives the Higgs field a non-zero vacuum expectation value. Wikipedia has more on the theory of the electroweak interaction.

One way to understand the relation between the force carrier's mass and the range is to consider the classical field theory corresponding to the massive bosons: The field equation for the classical field corresponding to the massive particles has solutions that decay exponentially away from a source on a scale inversely proportional to the mass (such a relation was first proposed by Yukawa who invented a theory of the nuclear forces based on then hypothetical massive scalar particles, similarly photons acquire an effective mass in superconductors and the electric field decays exponentially within superconductors).

The original theory of the beta decay proposed by Fermi, which is a precursor to the modern theory of the weak interaction, made the interaction a contact interaction between the fermions, so it was first modelled to have zero range.

Mandatory Disclaimer: The answer "why" the weak force has such a short range can't be answered finally, since we can only answer this question with reference theoretical models and nothing answers why nature is such that they describe it.