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HBS is much bigger. It has many more alumni and is a much older institution. With 1000 people graduating each year, it's natural that there will be more CEOs, world leaders, etc. Indeed, it does have better brand recognition amongst laymen.
Additionally, GSB only been regarded as a top b-school for the last 30 years, and it's has been only recognized as the best business school for the last 5-10 years.
However, if you look at the split between H/S dual admits, close to 70% choose Stanford. (HBS yield% is higher because of the larger class size, but Stanford wins the vast majority of dual admits).
In most prestige driven industries (PE/VC/Consulting), HBS is considered the "gold standard", but GSB is considered the "platinum standard". Stanford GSB has nearly half the admit rate of HBS, and is widely known to be more difficult institution to gain admission.
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