First they both involve water, Koi fish, plants and natural inhabitants. One of the big differences is where they get their water from, how it's treated or filtered.
A Koi pond is generally built with a pond liner to hold and maintain the water level.
A farm pond typically is built with the ground you walk on. The material needed is high clay content subsoil found below the topsoil. Using the earth to build the pond they are called earth pond and earthen dam. Even having enough clay content, water will seep a small amount like having small leaks but generally ok as long as the water supply can keep up with the leak. On the other hand finding you do not have enough clay content you will need a pond liner.
The major difference between the two is where the water comes from. With the Koi pond the water can be supplied by a well, city service, even trucked in for the bigger ones. Water treatment may need to be applied to remove the chlorine.
The farm pond water supply is added naturally from up to three different sources. These sources include underground springs and small seeps, part of a nearby stream can be diverted to the pond (with proper permits) and the watershed above the pond (rain). Typically the springs and seeps will be clean water, but check the water quality of the stream. The watershed should be a grassy field or a wooded lot, not a pasture for live stock or crops.
Another distinct difference is in their construction. The farm pond is below ground allowing the water to enter naturally from the top. The Koi pond is above ground where the water is added artificially. It is all in the terminology having the earth pond in a populated area is not a good idea.
Where is the water coming from? Do the neighbors have pets? Do they use fertilizer or other lawn enhancements? What else could be in their backyard that could end up in your pond? Therefore the above ground pond is the way to go if living in a populated area. Having a few acres in a less populated area you could have a below or earth ground pond, the farm pond.