Hi @Sven_BR_Lawyer and welcome to the forum!
If you use the project (i.e. non-Enterprise) export, it definitely will not; it only exports certain task information - no comments, change information, etc. (You can quickly do an export of a project and open the resulting file in Excel or Google Sheets to see what’s there.)
The Enterprise Organizational export does export a lot more data; you can see a specific list of what gets exported here. Note, though, that this Organizational export is not designed to be re-imported into Asana - there’s no facility for that. It’s more to know that you can get out your data and take it elsewhere.
The other option to explore is 3rd party services which export Asana data; a few I’ve heard of (though I’ve not personally used any of them so these are not endorsements) are Bridge24 and cloudHQ. I know probackup.io is also building one.