Question: What’s the difference between eating and drinking in regards to feeding your animal first where there is a mitzvah to give food to your animal before you eat but you don’t have to give a drink to your animal before you drink?
A. The Mishna Berura (167:6:40) says that if a person says a bracha, before taking a bite, besides being allowed to ask for the salt, or telling someone to pass a piece to someone, he can also say please feed the pet because there is a mitzvah to feed animals before you eat so if you forgot and said a bracha and about to eat then you can tell someone to feed the animals before he takes a bite. But in terms of drinking a person has the right to drink before his animals as we see that Rivka first gave water to Eliezer then the camels at the well.
B. There can be issues of getting too busy or suffering.
Answer: Footnote 31 in the Dirshu says in the name of the Ksav Sofer and Har Tzvi that a person will more easily get distracted while eating and forget to feed his animals then while drinking. Or a person is in more pain over thirst than over hunger so he has a right to drink first. The Ohr Hachaim adds that if he has a lot of hunger pains then he can eat first then feed animals.