It’s a difference in etymological origin. The etymology of “burgh” (with an “h”) is Scottish (the Scottish-English cognate of the English “borough”), while “burg” (without an “h”) is German. Between 1758 and 1890, both spellings of Pittsburgh/Pittsburg were used. In 1890, the federal government was trying to standardize place names, and they made the sweeping decision to drop all H’s from places ending in -burgh, so that they’d end in -burg. As to your point, the population of Pittsburgh in 1890 was 238,000 (page 4 of the Census Bulletin). In many non-federal applications, the “Pittsburgh” spelling was still used. In 1911 (population 533,000 - the downloaded zip file contains a file for PA), due to pressure from the people of Pittsburgh and one of the US senators from PA, the naming board reversed its decision.