Socks on their own I've found don't do much if your shoes aren't insulated, or you aren't using a warmer like those thermal foot warmers, or something else.
In lightly cold weather though, wool socks vs thin cotton socks can certainly make a difference. Once it drops to <40 degrees I find it's best to find better shoes, or use a warmer pad. I may look into insulated shoes myself, or get something more permanent like battery powered electric socks. I've heard people like stuff like this (warning: totally random example off Amazon) but I haven't done enough research or trial and error to decide yet. I'm thinking some insulated hiking shoes might work well combined with good socks for most cold weather.