A field does not always have to be used for farming. A farm is used for agricultural purposes.
A field can be a land area used to grow things such as vegetables, corn, wheat, and so on. It can also be used in terms of land, "an open field" means a section of land that is just grass. A farm is a place where a field can be, but it can also have animals on the farm. "On my farm I have cows, pigs, and a field of wheat." Farms will also usually have a house where the farmer and the family live, as well as some type of shed. If animals are kept on the farm it will have places to house the animals, like a barn or a stable.