Nutztiere are animals, which are used for economical reasons. Heimtiere are animals kept at home, but like cats and dogs not for economical reason, but for pleasure and company. The Gebrauchshunde are used for special purposes such as guiding the blind. Bauernhoftiere are animals used on a farm, mostly economically. The term Bauernhoftiere and Nutztiere have in common that both are used economically or for practical reasons, but the former live on a farm, while the latter could live somewhere else as well. Nutztiere are as well Lasttiere (animals used for carrying goods), Jagdhunde (dogs used for hunting) or Wachhunde (watch dogs).

I have never heard the word "Bauernhoftier" before, but its meaning is clear from its components: "Bauernhof" is a farm and "Tier" is an animal. So it stands for any animal you can find at a farm. The meaning of "Nutztier" can also be derived from its components: "Nutzen" (use) and again "Tier" (animal). So "Nutztier is any animal are of some use to humans (other than being a companion). Now it's obvious that the two concepts overlap quite a bit. But you might find a cat on a farm which usually is not a "Nutztier", and not every animal that is kept for a purpose lives on a farm (e.g. sled dogs or honey bees).