First off, let's make sure that we all understand what java.lang.String.intern() actually does... Basically, it internalizes strings such that for any two strings, s1.intern() == s2.intern() if and only if s1.equals(s2). In other words, it makes sure that the returned string reference points to the single, canonical version of that string of characters. It does this by managing a pool of unique string values.
Now, as noted in What are the differences between the == operator and the equals() method?, we know that, typically, we can't use the == operator to compare string values because the == operator is just comparing the values of references to those strings rather than the strings' values. But, for internalized strings, since we know that the value of the reference will always be to the unique, canonical version of the string, we can use the == operator.
Therefore, the primary benefit in this case is that using the == operator for internalized strings is a lot faster than use the equals() method. So, use the intern() method if you're going to be comparing strings more than a time or three.
The primary disadvantage is that you have to remember to make sure that you actually do intern() all of the strings that you're going to compare. [Note that you don't have to intern() string literals or string-valued constants.] It's easy to forget to intern() all strings and then you can get confusingly incorrect results. Also, for everyone's sake, please be sure to very clearly document that you're relying on the strings being internalized.
The second disadvantage if you decide to internalize strings is that the intern() method is relatively expensive. It has to manage the pool of unique strings so it does a fair bit of work (even if the string has already been internalized). So, be careful in your code design so that you e.g., intern() all appropriate strings on input so you don't have to worry about it anymore.