They are basically the same, but when I hear "I work at home" I think of a person who doesn't need to leave the house to do their work. Like a writer, or a computer programmer. When I hear "I work from home" I think of a person who leaves the house to do some of their work. Like a lawyer, or a plumber. So, if a person says "I'm a lawyer, I work from home" it gives me the feeling that they have a little office in their house but they also drive around the city to do their work. If they say "I'm a lawyer, I work at home", then I get the feeling that they stay home in their pajamas all day and work in their little office without much contact with the outside world. Furthermore, it wouldn't make sense for a plumber to say "I work at home", but it makes perfect sense for him or her to say "I work from home".

Depends on the context. In some contexts, they would be identical. On the other hand, "Work at home" can often refer to people whose sole workplace is their house. This is in contrast to "Work from home", which principally means your normal office/location of work is not your home, but you are working there for a period of time, ie. today, or this afternoon, or this week. 'Work at home' can often refer to the job of a housewife, wife, spouse (or husband, househusband), who does domestic work in the home.

Karl gave an excellent answer.

I think it has the same meaning..

same