The words are synonyms to a point, but "authenticate" is used more to mean that someone is proven to be genuine, or real, as in someone authenticating a painting.

Verify is used about anything, such as "I need to verify this schedule before I leave." The schedule is not being authenticated, because it is an actual schedule, but it simply needs to have the times and dates checked out.

  There is not much of a distinction to be made. The words   are synonyms.

http://www.thesaurus.com/browse/authenticate?s=t

The first sentence can use either word. The others cannot.

1) You must verify your email address. (Show truth of something to someone else)

2) This painting was authenticated by an expert. (Show truth of something that requires expert knowledge)

3) We cannot verify the date that the patient visited Africa. (Show truth of something that requires factual proof)

I agree with Rachel.

Authenticate = make sure or prove that an object is genuine.

Verify= make sure or prove that information is true.