Webster's definition of relish is "any of a variety of foods, as pickles, olives, piccalilli, or raw vegetables, served with a meal to add flavor or as an appetizer; or a pickled condiment, usually with ... spices, sugar, vinegar, etc."

A chutney is a relish made of fruits…spices, herbs and sugar with vinegar or lemon juice. As you can see this has bypassed the Indian origin of chutney.

The best answer to the question "What is the difference between a relish and a chutney?" is in O Chef  whose research lead them to "the generalisation that a chutney featuring a particular ingredient is going to be more savoury, tart and/or sour, than a relish featuring the same."