They are essentially identical in the example you've given.
However, "founded" has a bit weightier connotation.
It's not just that something was started at the time given, it's that the thing is a well-known institution.
It also needs to be an institution to be founded , while you can establish almost anything.
You could say Google was founded or established
You might say a law was established, but you would not say it was founded
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In literal terms: We found or establish companies or institutions.
We establish temporary guidelines, rules, protocols, and laws.
However, figuratively we can say: The law is founded on the principle of equal pay for equal work.
This policy relating to billable hours is founded on our firm conviction that work is far more important than family and free-time.
We expect our associates to work at least six days a week, and on Sundays whenever the situation requires.
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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 found ing 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.
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One way I think we could look at it, perhaps, is correlate 'found' with another related word 'foundation'.
Visualise foundation, and its connotation, as being that firm solid base on which further structures are built.
Imagine erecting a building without a foundation - That doesn't sound too assuring, does it??
Of course, the foundation, no matter how solid & long-lasting, is kind of meaningless, unless something is built on it.
Hence, TOOGAM's example of Redmond seems quite accurate.
I also agree with the fact that one difference between found & establish is in terms of 'permanence'.
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As I have seen some real world examples, founder means who generated an idea of any institute or organisation but the organisation is established by somebody else.
For example my father founded an organisation but the establishment of firm is done by me.
There might be a difference in the year of founding and the year of establishment.
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Google was  established  in 1998.  Google was  founded  in 1998.    In Merriam-Webster's Learner's Dictionary, both  establish (2a)  and  found (1)  share the definition below:   to begin or create (something that is meant to last for a long time).   They are especially confusing in the case of creating a company or organization, as in the example above indicate.  How do you differentiate them?
What's the difference between "found" and "establish"?
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<difference><verbs><comparison>
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They are essentially identical in the example you've given. You could say Google was founded or established. In literal terms:  We found or establish companies or institutions. We establish temporary guidelines, rules, protocols, and laws. However, figuratively we can say:   The law is founded on the principle of equal pay for equal work.  In the example they are essentially the same.
The "founded" connotation is however a little weightier.
In the example they are essentially the same.
The "founded" connotation is however a little weightier.
Not only did something began at that time, but it's a well-known institution.
It also needs to be founded and almost anything can be established.
You might say that Google was established or created
You could say that the law has been created, but that it has not been established. Literally we found or establish companies or institutions.
We have provisional guidelines, rules, protocols and legislation.
But figuratively speaking, we can say that the law is based on equal pay for equal work.
