I used to run outdoor orienteering constests in winter at a boarding school. The kids had to bring snowshoes, but whether they wore them was a team decision. At 2-3 inches no one wore snowshoes. At 8 inches almost everyone did.
The big difference for me was the ability of snowshoes to bridge things you couldn't see under the snow. Rocks and pocket gopher mounds in pasture and small logs in bush; hummock bogs, were the main things.
If on established trail, the boundary happened when you had to consciously lift your foot up to place it, rather than the 'just enough to clear the dirt' that is a normal walking pace.
Another determinate is the size of your group. Snowshoes are a win at shallower depths with a larger group. The first two people create a trail. Everyone behind them just walks on snowshoes.
On crusted snow, and frozen creeks, snowshoes give traction. And they help on elevator snow. (The snow that has a crust almost strong enough to hold your weight)
You can also get a situation where the crust will 'almost' support you on snowshoes. This often results in your tips getting caught under the crust. Sharply upturned wooden snowshoes made from separate side rails (Ojibway style) have less problem this way,
Note that on broken trail, snowshoes can be faster than walking. In our winter program the winning time on the 47 mile senior race was 12 hours and change, with most teams coming in in under 15 hours. (Boys, grades 10-12, teams of 5-7 that had to move as a unit, 1 hour stop for lunch, 30 minute stop for supper.)