[vaniotia]

To some this could be considered to be more of a physics question, but I think Chemistry is more accurate. It is a fairly simple question, and it is - how do the same subatomic building blocks, proton, neutron, and electron form an atom of an element having different properties depending on the quantity thereof? Is it some sort of holistic interaction with other particles? Disregarding specifics, it's sort of like saying 3 bricks makes water and 4 bricks makes metal, how is this possible? Yes, I'm honestly asking. A complete answer is not necessary, and indication of what field of research to pursue to find treatment of this subject will suffice, although the more thorough the explanation, the better.

[Klaynos]

I'm going to split my answer into two. The first is imo physics. "How do different nuclei form?" If we imagine that we can make any combination (which through processes that occur in nature supernova, nuclear fussion in stars and nuclear fission processes) all we need to know is how nucleons are bound together, and what makes them stable... This is quite an indepth area of physics, nuclear physics. From my memory I'd recommend reading about the liquid drop model for nuclei.

The second question is "what causes their different chemical interactions with each other?" This is fundamentally electromagnetic interactions, but is complicated by energy levels of electrons. A nucleus is positively charged, it therefore attracts electrons. What you need to investigate is how orbitals are formed around nuclei, and then how chemical reactions depend on the different electron layers, how filled they are, what their energies are etc...

[swansont]

How atoms bind with each other dictates the properties they have. So yes, you can have very different behavior by changing either the structure of the atom (changing the number of protons and therefore electrons) or by changing the bonds you have (which molecule you form)

[mississippichem]

I believe you are asking about the physics that gives rise to chemistry.

It's all about the behavior of the electrons in their orbitals surrounding the nucleus of the atom. Any individual electron is indistinguishable from any other electron but in atoms and molecules the electrons are organized into "orbitals" of different energy. The difference in energy comes from the properties that describe the "orbital" such as distance from the nucleus, angular momentum, spin etc.

Chemistry is really just the physics of the sharing of electrons between atoms. Electrons in these atomic orbitals are "stacked" in a predictable way governed by the laws of quantum mechanics. The number of electrons in the outermost energy level are the ones that really govern the chemical reactivity since only those are really available for sharing with other atoms.

This sharing of electrons is called chemical bonding.

Different atoms have different numbers of protons which governs the number of electrons needed to comprise a neutral atom. The way those electrons are arranged quantum mechanically governs how the atom will share electrons with other atoms and from molecules.