How are all atoms of iron different from the atoms of all other elements?

The most significant difference in the atoms of iron and atoms of other elements is the number of protons. The atomic number of iron is 26 and hence it has 26 protons. No atom of any other element will have the same number of protons. Similarly, iron atoms have 26 electrons and atoms of other elements cannot have 26 electrons (their ions might, not their atoms).

The same is not true for the number of neutrons, as the isotopes of iron may have the same number of neutrons as other atoms. For example, atoms of iron-54, iron-57 and iron-58 have the same number of neutrons as chromium, nickel and cobalt atoms (most abundant isotopes), respectively. 

In addition, most of the other elements do not have a magnetic character and hence iron atoms are different from most of the other atoms in that aspect as well.

Hope this helps.