'Mortal enemy' is a set phrase in the English language, and two 'mortal enemies' are two people who have hated each other for a long time, and will hate each other till they die (hence: mortal).
'Deadly enemy' is not a phrase I have ever heard before, but I think maybe it could be appropriate when talking about an enemy that can actually kill you. Example: When they started using the machine gun, the Germans became a deadly enemy to the British soldiers. OR In trench warfare, the deadliest enemy was often disease.
With this in mind, 'mortal enemies' is more appropriate, but still feels a bit too harsh for the situation. I would say 'sworn enemies' or 'they have had bad blood ever since...' (which means bad feelings towards one another due to an unsolved issue).
Espero que quede claro y avísame si hay alguna duda en español o inglés:)