Introduced is when the function was first written into the Mathematica function library. Updated is the last time the function has been changed. The introduced section is useful for users to know if their current version has access to that function. The updated section is mostly useful to let users know that the function will work differently between version and code you wrote in one version of Mathematica might not work the same way as it did in versions after those functions have been updated.
A good example of this is if you look at the FinancialData function that was updated in 2019. In the legacy version the output was in the form of a number, but in the newer version it is formatted as a dollar value (ex. $10). So if a user had code written before this update and ran it in a version after it was updated, the output would be different, and how you treat the results of the function would have to change. This is actually showcased in the documentation under the basic examples for the FinancialData function.
Integrate
experience many minor changes under the hood; I doubt that the documentation is updated each time when this happens. But, of course, I could be wrong. $\endgroup$ – Henrik Schumacher Apr 14 '20 at 16:12