I'm running both repair geometry in ArcPro and fix geometry in QGIS. In Arc I get a ton of Warning 000461: Repaired feature because of non simple and Warning 000986: > 25 non simple features. These are errors I'm not seeing in QGIS. What is Arc doing differently, and is it necessary?
If you have the original data, the Check Geometry tool should offer some more details on the specifics of each feature flagged, which would be captured in an output table when the tool runs. We built a reference document that explains the value of the output table here: https://pro.arcgis.com/en/pro-app/tool-reference/data-management/check-geometry.htm
To your question on necessity: When geometry data has non-simple features or topological errors, it can create unexpected behavior or errors in geographical software when viewing, analyzing, or editing data.
With that said, non-simple features may not need a fix to work with ArcGIS or other software. However, the tool is used to make features topologically valid under OGC or Esri standards, which helps to ensure high-quality spatial data which can be read by software using these standards thereby reducing the potential for geometry issues to arise later.