Why do electrons determine how atoms will behave?
We should be careful to specify the way in which electrons determine atomic behavior; for example, they don't really determine nuclear reactions or momentum. Their effect is largely relevant to the chemical reactivity of the atom.
Electrons determine this reactivity because, in simple terms, most atoms do not have a fully balanced charge. It's one thing to say that, for example, a hydrogen atom contains one positive charge and one negative charge, and is therefore neutral. However we also have to consider the fact that the electron cannot be located in a particular spot all the time, and the two charges are separated by some physical space. This means that there are inherent variations in the charge distribution of the atom (meaning, if you look at the same spot from one moment to the next, the amount of electrical charge will slightly change). We also have to look at the fact that electric charges diminish over distance, which means that if the electron is located on one side of the atom, the other side will experience less of a negative charge, meaning that no part of the atom is ever truly neutral. This leads to bonding, and it also means that even the noble gases are capable of forming compounds under the right circumstances.
Bonding and reactivity are products of this inherent variation. Although we have to remember that the atom isn't conscious, we like to say that it "wants" to behave in a certain way; that is, to have a full valence shell. This is because each successive shell places its electrons further from the nucleus, meaning that they experience less of the attractive positive force of the protons and are therefore more weakly attached. Electrons will always tend to go to where there is room in a valence shell for them to attach, and where the amount of positive charge they get to experience is stronger.
So, if we're just looking at electrons, then the behavior of two atoms will be based on the number of electrons they have, the space available for them, and the charge balance they will experience in those available spaces.