Niet is connected to horen. Niet is typically connected to verbs and adjectives, while geen is connected to nouns. To use geen, the sentence would have to be 'Wij horen geen dieren. 'De' in the original sentence makes the difference as it is specific and requires niet to negate the sentence. It would be the difference between "We do not hear the animals" and "We hear no animals"
I believe that it is just a different stress. Negate the whole thought or negate the object specifically, which would also make sense to me. When you negate the animals specifically, you are saying you do hear other things. When you negate the whole thought, we don't know if it is just the animals or if there is some noise preventing you from hearing the animals that might also prevent you from hearing something else or if we are deaf and wouldn't hear any way. I think both are viable options, but in English we would tend to say it the same way. Although if you really wanted to stress the animals were the only thing we didn't hear, I suppose we could say "We hear, but not the animals." but that would be again another translation in Dutch. Here is the grammar link for niet and geen: https://www.duolingo.com/comment/3734833
Here is the list of grammar explanations: https://www.duolingo.com/comment/3732817
To negate the animals specifically in Dutch, they would use "geen" instead of "niet". "Wij horen geen dieren." would be "We hear no animals." If you want to talk about specific animals "We do not hear the animals.", you could place "niet" as Jun-Dai suggested "Wij horen niet de dieren."
The sentence above "Wij horen de dieren niet." negates the whole thought including the verb. In English, the stress would be on the verb to negate it, or for the whole thought on the negation. "We don't hear the animals." or "We do not hear the animals."