Will my Apple Watch have a different phone number from my iPhone

This depends upon your carrier. If you Set up an Apple Watch for a family member - Apple Support (Family Sharing setup), then yes. If not, and if the Apple Watch is cellular, you will have to enable number sharing through your carrier. if the Apple Watch does not have cellular, then your phone and Watch will by default have the same phone number.

Assuming cellular Watch… All devices connected to a cellular network will have a unique number assigned by the carrier, so technically there are unique numbers for each. With Apple Watch, the carrier will typically route calls to the iPhone number to ring on both numbers (when the Watch is out if range of the paired iPhone and has thus its cellular radio powered up), so you won’t see or even use the Watch number in most contexts.