The Apple Weather App uses data from the Weather Channel. There will be any difference in the data displayed on your watch. You don't have to use the built in Weather app, but if you want weather data from a complication on your watch face, you won't be able to use something else for that purpose. If you're OK with that, use any weather app you want, which can be downloaded to your watch. I also have Weather Bug on my Watch.
Edit: In checking further, it appears the Weather Channel (which is not from Apple) has not released their app for Apple Watch. So you can't use that app. But again, the data for weather on your watch does indeed come from Weather Channel.