How have you configured the labels? Are you:
- Using styles?
- Using Arcade?
- Using custom fonts?
- Or is it simple, default labeling?
It's likely that you have used label styles or Arcade functions that are not supported in the ArcGIS API for JavaScript 3.x, which Map Viewer Classic and the Web AppBuilder run on. The new Web Map Viewer and newer ArcGIS apps use the ArcGIS API for JavaScript 4.x.
Here's an example article that covers a scenario where the label styles are not supported.
https://support.esri.com/en/technical-article/000027449
It also links out to Apps using 3.x vs 4.x for more information:
https://community.esri.com/t5/arcgis-api-for-javascript-documents/apps-that-use-arcgis-api-for-javas...
If you need more help from the community, please share how the labels were configured.
Cheers.