What enables protons to give new properties to an atom every time one is added?
You are not correct in your latter part of the analysis; the chemical properties (which is mostly what matters in ordinary matter) almost only depend on the electron shell, and in particular the outermost electrons (called the valence electrons).
So more protons mean more electrons and a different electron shell, meaning different chemical properties.
Why there is such a diversity of properties just by changing around the electron shell, is one of the wonders of chemistry! Due to quantum mechanics, the electrons don't simply spin around the nucleus like planets around the sun, but arrange themselves in particular, complicated patterns. By having different patterns, you can achieve a lot of different atom<->atom binding geometries, at a lot of different energies. This is what gives the diversity of chemical properties of matter (see the periodic table).
You can add or remove electrons to an atom to make the electron shells look more like the shells of another atom (with a different number of protons), but then the atom as a whole is then no longer electrically neutral, and due to the strength of the electromagnetic force, the resulting ion does not imitate the other atom type very well (I'm not a chemist - I'm sure there are properties that indeed could become similar).
Many physical properties are also mostly due to the electron shells, like photon interactions including color. Mass obviously is almost only due to the nucleus though, and I should add that in many chemical processes the mass of the atoms are important for the dynamics of processes, even if it isn't directly related to the chemical bindings.
This was just a small introduction to chemistry and nuclear physics ;)
Also, it is obvious that adding (or subtracting) electrons does not make a difference [...]
The two differences you describe between copper and zinc are in fact due to the electrons in the atoms. So the crucial difference between the two atoms is that they have different electron configurations in the electrically neutral state (when the number of electrons equals the number of protons).
The different colors are due to particular wavelengths of light that the electrons emit when they switch from an excited state back to their ground states. So different electron configurations lead to the different colors. Similarly, conductivity depends on having (almost) free electrons in the metal such that they can form an electron gas. If this is the case crucially depends on the electrons in the outermost orbits of the atom.
Ions of the same elements behave completely different. You just cannot have a stable lump of (say) copper ions with a color and a conductivity.
There are two processes when you add a new proton to the nucleus, aiming to get a new neutral atom:
- Addition of a proton, which increases nuclear charge by 1
- Addition of an electron, which compensates the nuclear charge increase to make the atom electrically neutral
Let's consider these two parts of the process separately. First, suppose you have an atom of californium, which has 98 protons. If you remove an electron, you'll get an ion, energy states of which will highly resemble those of berkelium, with one major difference: $\mathrm{Cf}^+$ is no longer an electrically neutral object. This means that even though single ion will behave much like $\mathrm{Bk}$ atom (i.e. its radiation and absorption spectra, spectral line intensities will be qualitatively very similar), it's not stable with respect to addition of an electron. This affects its chemical properties. The ion "wants" to acquire back its missing electron, and when it sees an atom, it tries to attract one of its electrons. The same goes further: when you remove a second electron, getting $\mathrm{Cf}^{++}$, this ion will have very similar properties to those of $\mathrm{Cm}$, but it will not be stable with respect to addition of electron, and thus interaction with other atoms and ions will also be much different.
Now what can we do to fix this instability? Of course, we have to compensate the increased total charge by removing a proton from the nucleus. After you remove a proton (and corresponding number of neutrons to prevent fission) from e.g. $\mathrm{Cf}^+$'s nucleus, you'll get nothing but already familiar $\mathrm{Bk}$ atom, which will be electrically neutral and thus not tending to acquire extra electron (at least not tending that much — bound states with extra electrons are always possible).
What is the conclusion? Simple: adding another proton to the nucleus just selects stable electron configuration from all the possible configurations. And the "default" configuration selected is what makes the atoms have so different chemical properties.