I don't follow fossils, but it's my understanding that lots of fossils are virtually the same as animals alive today. But 5000 years is lots of time for speciation to make little changes. And of course, a lot have gone extinct.
I happen to be a geologist published in paleontology of invertebrates. Fossils of the fossil record become more and more different the further in time we go back.
Animals of the pleistocene for example, in some cases are like animals today, such as the case of a whooly mammoth or sabertooth tiger. These animals are similar to elephants and similar to tigers and lions. We also had giant armadillos and giant sloths and giant bears. These sort of similar to what is alive today, yet different.
If we go back into the early cenozoic, whales transition, going backwards, back into land animals. Horses have multiple toes, elephants become more like pig sized without tusks, etc. A lot of animals change.
Some animals remain somewhat similar, such as alligators and sharks. Though as we know, megalodon and prehistoric alligators were some 30+ foot long, far larger than sharks and alligators today. There were titanaboas as well.
Shellfish and starfish, and other species of fish remain similar yet slightly different over time.
As we go back into the mesozoic, of course we have dinosaurs, unlike anything that is alive today (except for birds in that they share talons and feathers, their hip bones are somewhat similar as well, and the shape and details of their skills.
Mammals were largely different. We had anapsids and synapsids, mammal-reptile hybrids, unlike anything today. Flying reptiles such as pterodactyls and swimming reptiles such as pleisiosaurs. Also unlike anything today. Shellfish and some fish and animals of the deep seas still changing yet slowly and somewhat similar to those today.
And as we reach back into the Paleozoic, things stretch back and change even more. Of course fish with toes and wrist bones were present by the early devonian. In the earlier Paleozoic up until the silurian there were no land animals. Which obviously is different from today. Prior to carboniferous there were no reptiles or birds or mammals at all, in the sea or on land. Much different than today. Though after the carboniferous reptiles began to dominate, still long before mammals and birds appeared.
And as we stretch back further to the early Cambrian, all we have to do is Google "opabinia" and "anomalocaris", and within a matter of seconds we can see that life was significantly different then than today. And even as we go back further into the pre Cambrian, we have microscopic shelled animals like cloudina and sinotubulites. We have worm-arthropod hybrids and contested species of arthropod, mollusks and annelids. Animals that are difficult to classify because they hold traits of multiple genus'. They being soft bodied are also few in number. But again, very very different than modern day life.
So the point is that, when we really spread the geologic column out, we certainly see continual change over time, and no modern day species is found anywhere in the fossil record beyond the latest parts of the cenozoic.