There may not be any difference - working from home implies that home is a base and you will work in other places as well, including home - for example a sales person my do their paperwork , research potential customers at home, but their main time is visiting customers at their site.
Working at home implies that most of the work is done in your home, for example somebody who puts together mailshots, or rings around to get appointments for sales people, does it from home.
I have done some work at home but this was not the norm...............however when I was working from home for 2 years my office was in my house, and had to qualify for HSE checks and insurance.
Work at home and work from home are nothing but two different terminologies of home based work businesses. Home-based work gives you the freedom and independence to work from your home at your leisure. Working from home gives you the scope to become the boss of your own company.