The Halacha states that one is prohibited to sit down to eat before he feeds his animals. (Orech Chaim 167;6) This is derived from the Pasuk we recite twice daily in the second paragraph of the Shema. Hashem promises He will provide grass in the fields for the animals and you will eat and be satisfied. Since Hashem first mentions He will feed the animals and then man hence we should follow the same practice.
With regard to drink this rule does not apply. Man should drink first and then give the animals to drink. This we cans see in our Torah Portion of the week. When Moshe is instructed to extract water from the rock, Hashem says to Moshe: VEHISHKITA ET HA’EDAH VE’ET BE’IRAM, “…and you will give drink to the assembly and to their animals.”(Num 20,8) Here we see that when it comes to drink people come before animals.
This is also seen even more forcefully from Rivka. When Avraham’s servant asked her for water she first gave it to him and then offered to give water to his camels.
Perhaps the reason for the difference between food and water is that a person can go without food a longer period of time than he can without water.