The Wellness app for the smartphone is, as said above, an Asus app. It does NOT count steps on the phone, but only syncs with the watch and therefore only shows you the steps that the watch measured.
Google Fit on the other hand was primarily designed for phones and uses its sensors to gather steps and other stuff. In addition, Google Fit can also gather the data from Wear Smartwatches (and other 3rd party wearable devices). In the Android Wear app on your phone you should find a setting where you can configure which app should be used on the watch for steps and heart rate. If you select "Wellness" there, Google Fit will not receive any data from the watch, only the Wellness app will. If you select "Google Fit", it's vice versa.
Usually it makes more sense to use Google Fit, because when you do, your steps will be counted if you have EITHER your watch OR your phone with you. And Google Fit is smart enough to not double-count the steps if you carry both. With Wellness on the other hand, you have to wear the watch in order to count your steps.