How can two atoms have the same mass but different charges?

The answer is: two atoms have same number of protons and neutrons, but different number of electrons.

Mass number (A) is the total number of protons and neutrons in a nucleus.

Atomic mass (ma) is approximately equivalent to the number of protons and neutrons in the atom.

For example, chlorine atom (Cl) and chlorine anion (Cl⁻).

Similarities:

1) atomic number is the same. Both, chlorine atom and anion, have atomic number 17. They have 17 protons in nucleus of an atom.

2) both are the same element.

Difference:  

1) chlorine atom (Cl) has neutral change and chlorine anion (chloride ion Cl⁻) has negative charge, because chlorine anion has one electron more.