On my ArcGIS WebMap the address point are styled correctly (orange with labels in the first screenshot), but when I view the webmap in a webapp they are not (they are grey with no labels in the second screenshot). What might cause this issue?
You are using the newer map viewer which uses javascript 4.x, while Web AppBuilder uses the retired javascript 3.x. You would need to build your maps in Map Viewer Classic to ensure full compatibility with Web AppBuilder.
Note that Map Viewer Classic and Web AppBuilder are both retiring this year. Map Viewer Classic Deprecation Notice and Experience Builder Retirement Notice
I recommend moving to Experience Builder, as it is fully supported and compatible with your current map. Migration Resources for ArcGIS Web AppBuilder
Try either creating your maps from ArcGis pro and then deploying them on arcgis online; especially visit https://community.esri.com/t5/arcgis-experience-builder-documents/migration-resources-for-arcgis-web...
you also need to check your base layer labels very carefully before deploying the map or database.