This is a great question. The main scientific difference is that they come from different branches of the crustacean family tree. They both have 10 legs (members of the decapod order - meaning 10-footed). However they come from different suborders. Shrimp are pleocyemata along with lobsters and crabs, but prawns are dendrebranchiata.
Both of these organisms are thin exoskeletons but in shrimp the thorax overlaps the head and abdomen and in prawns each part overlaps the part below it. This means that prawns cannot move their bodies as sharp as shrimp.
Prawns have 3 pairs of legs while shrimp have only one. Along with this they reproduce differently. Prawns release their eggs into the water to grow independently, but shrimp carry their fertilised eggs in their bodies.
It can be quite confusing because in the UK, Australia, New Zealand and Ireland we call both true prawns and shrimps, 'prawns'. In North America the word 'shrimp' is used more but 'prawns is used to identify larger species of these fished in fresh water. This can make it difficult to know which one you are purchasing (if purchasing at all) I know that I would certainly get confused.It also makes me wonder we don't just call them by their true names, consistently.
I hope this helps.
A Giant Tiger Prawn