What is the difference between a predicate and function?

A predicate is a box that takes an argument and returns a Boolean value. For example, "$x \mapsto x \text{ is even}$".

A function is a box that takes an argument and returns a value. For example, "$x \mapsto x^2$".

Edit (following Amy's suggestions): There is some domain over which all variables range. A function takes zero or more arguments of that domain and returns another argument from that domain. A predicate takes zero or more arguments of that domain and returns a Boolean value.

The terms "Function" and "Predicate" are only and solely determined by the formal system in which those words are being used/defined. In most formalizations of first order predicate logic the words "predicate" and "function" classify two different kinds of signs (the function signs and the predicate signs) each of which follow different rules of use i.e. function signs are different from predicate signs because the rules for using function signs are different from the rules for using predicate signs.