Having the same number of neutrons Sorry if this is a silly question.
If I understand correctly, for two atoms "having the same number of protons" is equivalent to "being of the same element", while "having the same number of protons and the same number of neutrons" equates to "being of the same isotope (of the same element)".
But does "having the same number of neutrons" in itself have some significance in physics? And what about "having the same total number of protons and neutrons (but not necessarily with the same summands)"?
 A: Two nuclides (a type of a nucleus with some values of $Z$ and $N$) that have the same number of neutrons $N$ have the same number of neutrons. The previous sentence is a tautology but I wrote it to show that the equality doesn't mean anything special, aside from the things that are obviously implied by it.
Two nuclides that have the same total number of nucleons - neutrons plus protons - have the same total number of nucleons - neutrons and protons. In the latter case, it also means that the two nuclides are approximately equally heavy and that there may be a beta-decay transformation from one to the other (beta decay doesn't emit any protons or neutrons, so it approximately conserves the mass of the nuclide).
In both cases - equal $N$ or equal $A=Z+N$ - are less important than if two nuclides have the same $Z$. The number of protons $Z$ is by far the most important quantity that determines the character of the resulting material. It's because the electrons only feel the electrostatic force of the nucleus and it only depends on $Z$; it doesn't depend on $N$ or $A=Z+N$. A large chunk of matter has to neutralize the electric charge, so it has to assign the right number of electrons - namely $Z$ electrons for one nucleus.
Consequently, these $Z$ electrons get inevitably arranged into the shells of atomic physics and the properties of the material are therefore a quasi-periodic function of $Z$, the number of protons, as the periodic table makes very clear. That's why the chemical properties of e.g. iodium-127 and iodium-131 are almost identical while iodine-131 and xenon-131 are extremely different chemically, despite their having the same value of $A=N+Z$.
Of course, the mass of the materials depends on $A=Z+N$ and not just $Z$. Also, some isotopes are radioactive while others are not. But radioactivity is not really "chemistry" or "material science"; radioactivity is a rare process that only influences the nuclei while chemistry and material science depends on the electron shells.
A: Isotones are nuclides having the same number of neutrons. 
Magic proton or neutron numbers give the nucleus greater stability. Magic 82-isotone nuclides for instance:

Isobars are nuclides having the same mass number (i.e. sum of protons plus neutrons). 
The number of protons in beta-plus (beta-minus) decay decreases (increases) by a unit and the number of neutrons increases (decreases) by a unit, so that an isobar standing to the left (right) of the original nucleus is formed.
It may be 1, 2 or 3 beta-decay stable isobars.  Beta-decay energy of 154-isobar nuclides for instance:

Isotones and isobars have great significance for studying  of nuclide stability.
