Why are the sales and finance departments each other's worst enemies???

Because they depend to each others !!

Sales and Finance department are pillars of businesses empire.
Sales team will always quote/think that they are no.1 and finance team will quote/think that without their sanctions sales team are handicapped.
This is because Sales is abided by business owner and the business policies are softcorner to sales but the Finance is abided by business policies and policies are great hand for good  businesses.
When two strong teams meat each other they will always criticise each other, it is humanbeings habit and nature.
Therefore, everyone will always feel that sales and finance departments are wrost enemies.
In a fact finance team is always good team to follow and act to build good business empire.

In simple meanings : once u have no trust , u will have anemies,Means sales they believe that more expenses more sales & profit,meanwhile finance believe that such expenses unvaluable cost can be avoided.