What is the most important reason why two country's with the same GDP can have vastly different standards of living?

As with individuals, nations have different amounts of wealth, and even if the wealth is fairly distributed, there will and is considerable differences in the standards of living among the several hundred nations in the world. As mentioned, regarding fairness of distribution, the standards of living can be different based on non subjective reasons.

A nation that is rich in Natural Resources and can export these as well as use them internally to their best advantage will help lift the standard of living.

A nation with a sound economic system will also help its standard of living. Another factor is the education level of a nations citizens. Generally the higher the level the higher the standard of living.

A politically stable nation is best for a sound economy. Nations in the midst of civil strife and civil wars hurts its economy and helps lower its standard of living.