Weatherby is noticeably lighter and a gas operated gun. I find its length of pull is shorter and fits me better.
Stoeger is using a copy of the Benelli inertia system, it is not a Benelli for 1/3 of the price the one I had encounters with had issues but they were operator caused. The owner never cleaned it properly, it became a single shot as soon as it started to rain.
If I wanted a low cost recoil operated gun the Franchi Affinity is what I would look at but the Weatherby with what appears to be a copy of the Beretta gas system and its light weight and much lower price point is the one I will probably be getting next.
At the end of the day between the Stoeger and Weatherby I would buy the one that fits me best and most likely never regret it.