The Apple Watch will stay paired with your phone when resetting all settings.
Reset All Settings basically resets all settings and leaves your data alone (turns off and forgets your location services settings, forgets your notification settings, forgets your privacy settings, deletes your Apple Pay cards, etc). You'll stay signed into iCloud though. (Personally, I think it can be a bit of a pain having to set up all the settings again. Most apps are good about asking if you want to use location services or get notifications again...but some don't, so then you have to delete and reinstall the app to get the prompt.)
A factory reset would be completely wiping your phone, and deleting your data. You'd either have to set up as new or restore from a backup.
Usually when troubleshooting, you'd do the Reset All Settings first...and if that didn't fix the problem, then do the factory reset. I'm surprised Apple is having you do it the other way around.
On iPhone, return settings to their defaults without erasing your content.
support.apple.com