As a new parent, you'll have gotten used to the subtle differences in your baby's cry meaning that they're wanting different things, and an overtired baby cry is no different. Generally speaking, overtired babies will have a cry that is more high-pitched and nasally than other cries, and it might be accompanied by jerking motions or an arched back.
As you can imagine, this only stresses out your baby further, and isn't ideal when you're wanting them to fall asleep and stay asleep through the night!