"This is an exact number."
Translation:C'est un nombre exact.
In some cases they are interchangeable, but "nombre" refers to a count or amount, while "numéro" just refers to a sequence of digits, like a telephone number. Note that a house number is a "numéro", because it's not actually about counting houses, but about an arbitrary group of digits used as a label.
In this case, "numéro" doesn't make much sense, because if you're questioning its exactness, it can be an approximation. Thus, it's probably a count or measurement of something.