https://api.inaturalist.org/v1/docs/#!/Observations/get_observations
See also here for more details on some of the parameters.
Many, but not all of the API parameters are implemented on the Export page. So if you build a query using the API docs, you may get something different when you plug it into the Export page. On the other hand, If you copy the parameters from the box on the Export page to an API call, you should get the same results.
Compare https://api.inaturalist.org/v1/observations?quality_grade=any&identifications=any&place_id=97394&taxon_id=459050&verifiable=true to your Export page screenshot.