In normal usage, there is no difference.

However, they are nonetheless slightly different.  

A coach is slightly more focused on the mental approach of the person.  (A sports coach will focus a little more on strategy and tactics.)  

A trainer is slightly more focused on the skill of the person.  (A sports trainer will focus a little more on strength and agility.)  

An elite sportsperson would have one coach, but may have more than one trainer, each focusing on a particular skill.