The latter explicitly creates a new and referentially distinct instance of a String object; the former may reuse an instance from the string pool if one is available.
You very rarely would ever want to use the new String(anotherString) constructor. From the API:
String(String original): Initializes a newly createdStringobject so that it represents the same sequence of characters as the argument; in other words, the newly created string is a copy of the argument string. Unless an explicit copy of original is needed, use of this constructor is unnecessary since strings are immutable.