What do you call a scientist who studies fossils?

A palaeontologist. The distinction people get wrong is with archaeology: archaeologists study human remains and human things, palaeontologists study everything else and generally work on timescales where humans do not appear at all. Nobody digs up a dinosaur and a Roman villa in the same trench.

The field was substantially built by Mary Anning, who from the age of about twelve collected fossils from the crumbling cliffs at Lyme Regis to support her family. She found the first correctly identified ichthyosaur, the first two plesiosaurs, and a British pterosaur, and worked out that the stones called bezoars were fossilised faeces containing the animal's last meal. As a working-class woman she could not join the Geological Society, and gentlemen published her finds under their own names.

Most of what a fossil records is not the animal. Bone is usually replaced mineral by mineral until nothing original remains - a fossil is a stone copy. Only in exceptional conditions is anything else preserved, which is why sites like the Burgess Shale, where soft bodies were captured, matter out of all proportion to their size.