What's the difference between mass number and atomic number?

Atomic number equals the number of protons (positively charged) in a nucleus, while the mass number (AKA "atomic weight" or "atomic mass") is the number of protons plus neutrons.  In a neutrally charged atom, protons and electrons (negatively charged) will number the same.

They've started using "mass number" to differentiate it from the average "atomic mass" gotten by averaging all the isotopic weights for an element.

if you look at the periodic table, the elements are numbered from 1- whatever. the single, whole number is the atomic number. it is the number of protons and you can also say that it is the number of neutrons. the atomic mass is (usually) the larger number which has a decimal. that is the number of protons AND neutrons combined. to find the number of neutrons you subtract the atomic mass - the atomic number.
example)
Say you are trying to find the number of protons, electrons and neutrons in Carbon (C).
the atomic number is 6. it has 6 protons and therefore it has 6 electrons.
the atomic mass is 12.01. so you should subtract 12.01 (mass) - 6 (atomic number) to get  6.01 or just simply 6 neutrons.

Usually the number of neutrons is the same, but for certain elements they are different. This is called isotopes