Hearing would still be a thing, since I'm assuming animals would still produce sounds by moving around; also, enviromental sounds would be still relevant to everyday life in the wild. After all, most predators aren't exactly used to alert their prey with unnecessary noises (compare this with how most felines sneak around before pouncing on the prey, and how wolves howl to communicate between the pack, certainly not with the prey).
It would be interesting to determine how human can speak, while animals do not. From an evolutionary standpoint, if no one has ever developed vocal chords before, it's difficult to imagine them come up with the "homo" genus.
Maybe there is an additional link in your human evolution, bringing the "homo loquentes" (speaking man) somewhere between the austolopitecus and the homo habilis (developing vocal chords and the brain power to use them is no small task; also, that brain power would then come in handy for the "habilis" species; but I guess you could as well place the loquentis between habilis and erectus, if you prefer).
But what about our ancestors? Chimps in our world use both verbal and non-verbal communication. In yours, they would use only the latter (meaning facial expressions, gestures, stances, and so on). We can still imagine sound being used to some extent (like, tapping a foot or a finger against an object to refer to it, or deliberately making a fuss to signify anger).
Since non-verbal communication would play a major role in all the animal world, to compensate for the lack of voice, we can assume that we humans would have this, too.
So your variety of homo sapiens sapiens would be able to talk like we do, but I suspect they would rely even more on non verbal cues. The part of the brain responsible for recognizing facial expressions and making them would be even bigger, and we may have even a greater control of our facial muscles (e.g., being able to move our ears, or our eyebrows to a greater extent, would be common skills). Posture and hand gestures would play a bigger role in day-to-day life.
Psychologically, we would probably dislike strong noises, like shouting out, considering them even more rude. Moreover, maybe we would feel even more different from animals, the homo genus being the only one capable of speech and verbalization. Creationists would have another argument under their belt, pointing out how animals cannot speak, nor produce sounds.
Studying animal behaviour would be more difficult.