I built a few dozen homes and only once had a client request 2x6 construction. I tried to sell it on all the others, citing sturdiness and better insulating properties, but everyone looked at the features it excluded and went for flimsy 2x4 builds.
It gets important if you are a developer building subdivisions of homes. $10k saved on every house is $big profit. 25 home subs is $250k+ savings...very much an issue.
My own house is 2x6, and properly insulated top to bottom. My neighbor's slap job summer cooling bill is not quite double mine. Close to same sqft and same interior ac temp setting.