First of all, the word "found" has multiple definitions/meanings, including the definition of being the past tense of the word "find". I will ignore that as your question demonstrated your desired focus.
To "found" something like an organization or a city basically means to begin it. If I start a city, giving it a name and getting it to be legally recognized by a larger government, then I founded the city.
However, the term "establish" could also refer to the point when something became viable, longlasting, or true.
For example, there was once a city that was created on the last day of the year 1912. In the 1920s, economic hard times led to closing businesses including lumber mills. In the 1980s, the city had a strategy of providing nice land for cheap, hoping to attract some businesses to move to the city. The strategy worked; in 1982, they attracted a Japanese company enough to place their American headquarters in the city. In 1986, they also attracted a company whose product was a newfangled thing called computer software.
Although the city was formally founded in 1912, which was the day that this place became established as a city, it was really the investments in the 1980s by Nintendo and Microsoft that led to Redmond, WA becoming established as a technical hot spot with significant worldwide influence.
Maybe the city of Bellingham, WA had rosier prospects to have significant economic impact in 1911 (as Bellingham had already been founded in 1904), now now Redmond's role as an economic powerhouse has been firmly established.
Basically, "founded" refers to the start of something rather official, while "established" refers to the beginning of something that has remained, and something can be established over a longer period of time.
Here's another example, showing events that happened over periods of years: e.g., the United States of America's founding fathers lived in the late 1700s, but it was during World War 2 that America established itself as a worldwide leader known as a superpower.