The mass number is defined as the NUMBER of protons and neutrons. Electrons aside, why does the number of protons and neutrons equal the atomic mass rounded to the nearest whole number? That would mean, to me, that the number of protons and neutrons equals almost exactly to their relative mass! Why is that true?
This is incorrect. The atomic mass you read off the periodic table is the weighted average of all isotopic masses. For the most part, the masses of the isotopes are just the sum of the protons and neutrons, but nuclear binding energy causes some deviation. Also, protons and neutrons don't have the same mass, so for larger atoms, that will also cause deviation from a simple sum.
The biggest issue like I said is that atomic mass is the weighted average. For example, the two most prevalent isotopes of bromine are Br-79 and Br-81, present in about the same relative abundance. However, the atomic mass is 79.9 because it is the average. That rounds to neither 79 or 81.