The classic era of United States ("US") commemorative coins includes several issues that mark anniversaries of early municipal milestones. Three milestones typically have been commemorated: City Charter grants, foundings and incorporations.
Put simply, an incorporated city, is one that it is a local, self-governing entity with elected officials. A city's founding is more an indication of a permanent settlement. Settlements can develop into incorporated cities as they grow/develop.
Here's a quick look at three coins that are often mistaken as commemorating a city's founding date though they actually mark a later date of a city's incorporation.
1. Bridgeport, CT: The 1936 Bridgeport coins marked the 100th Anniversary of the incorporation of the city of Bridgeport, Connecticut in 1836. It can trace its initial (European) settlement (i.e., founding) to the 1600s, however.

2. Cleveland, OH: The 1936 Cleveland coins were struck in commemoration of the centennial anniversary of the 1836 incorporation of the City of Cleveland, Ohio. It was founded, however, decades before in July 1796.

3. Hudson, NY: The 1935 half dollars, per their authorizing legislation, were struck to commemorate the 150th Anniversary of the founding the city of Hudson, New York in 1785. Such date is actually its incorporation date, however; it was founded in 1783 as Claverack Landing.

Those are three members of the classic-era US commemorative coin series that many collectors mistakenly assume to be marking a city's founding when they are actually commemorating its incorporation.
For other of my posts about commemorative coins and medals, including more Quick Bits stories and others about each of the coins referenced here, see:
Commems Collection.