Difference between saving and exporting
AS @lhf comments export/import implies that there is some information lost during one of the processes so that you can't export and then import and get exactly the same document back.

Save usually writes the file in the format that is native to the application or in the same format as the file was opened in.

Some applications e.g. Microsoft Office will not use export as much as this but use save as to allow conversion of the format of the document and possibly lose some information e.g. Excel can save as .csv. (I don't have MS Office here and can't remember if they have an export on the file menu). The do have an import that adds the selected document to the current one so another meaning.

The examples you give are both that the file file format is native to an application ( Open Office and PhotoShop) and the other applications' export converts their own format to these on export.