The chief difference between a milligram and an International Unit is that a milligram is a measure of mass, whereas an International Unit measures the potency of a substance being used in a medical context. In addition, milligrams are part of the metric system and International Units are not.
International Units quantify the potency of biologically active substances, such as medication, vitamins, and vaccines. There is no single, set International Unit -- it is merely an agreed-upon standard for each individual substance. For example, an International Unit of vitamin A has a very different mass from an International Unit of vitamin C, because the two substances differ in their potency.
Milligrams, on the other hand, measure mass. They are a subunit of the kilogram, the standard metric unit of mass. Milligrams do not differ from substance to substance, as International Units do: a milligram of vitamin A is identical to a milligram of vitamin C.
International Units for a given substance are determined by the World Health Organization's Expert Committee on Biological Standardization. Milligrams are based on the kilogram, and do not have a governing committee.