How do the differences in the number of protons result in such great differences in elemental properties?
Unfortunately, I have to disagree with trb456's answer.  It is both the protons and the electrons that make up chemistry. (Neutrons are considered massive "spectators" though they are what keep protons (all + charged) together)

Bonds are formed due to the attraction of electrons and protons. (cf. Slater, Bader, et al.) (The precise shape and nature of the electron density is additionally determined by constraints such as Pauli Exclusion and electron-electron repulsion.)

So if you think about it, there is a very complex interplay upon mixing electrons up with protons. More positive charge in the nucleus tends to keep electrons drawn tight, but in some ways that is counter-balanced by electron-electron repulsion--and the fact that electrons are attacted to other nuclei as well.

So although we ascribe color ("electronic transitions") and reactivity to electrons, this is only part of the story. The protons are also there, and they provide some governance (via Coulombic interaction) over what electrons are permitted to do.