From JavaNotes:

• a.compareTo(b):
  Comparable interface : Compares values and returns an int which tells if the values compare less than, equal, or greater than.
  If your class objects have a natural order, implement the Comparable<T> interface and define this method. All Java classes that have a natural ordering implement Comparable<T> - Example: String, wrapper classes, BigInteger

• compare(a, b):
  Comparator interface : Compares values of two objects. This is implemented as part of the Comparator<T> interface, and the typical use is to define one or more small utility classes that implement this, to pass to methods such as sort() or for use by sorting data structures such as TreeMap and TreeSet. You might want to create a Comparator object for the following:

  • Multiple comparisons. To provide several different ways to sort something. For example, you might want to sort a Person class by name, ID, age, height, ... You would define a Comparator for each of these to pass to the sort() method.
  • System class To provide comparison methods for classes that you have no control over. For example, you could define a Comparator for Strings that compared them by length.
  • Strategy pattern To implement a Strategy pattern, which is a situation where you want to represent an algorithm as an object that you can pass as a parameter, save in a data structure, etc.

If your class objects have one natural sorting order, you may not need compare().

Summary from http://www.digizol.com/2008/07/java-sorting-comparator-vs-comparable.html

Comparable
A comparable object is capable of comparing itself with another object.

Comparator
A comparator object is capable of comparing two different objects. The class is not comparing its instances, but some other class’s instances.

Use case contexts:

Comparable interface

The equals method and == and != operators test for equality/inequality, but do not provide a way to test for relative values.
Some classes (eg, String and other classes with a natural ordering) implement the Comparable<T> interface, which defines a compareTo() method.
You will want to implement Comparable<T> in your class if you want to use it with Collections.sort() or Arrays.sort() methods.

Defining a Comparator object

You can create Comparators to sort any arbitrary way for any class.
For example, the String class defines the CASE_INSENSITIVE_ORDER comparator.

The difference between the two approaches can be linked to the notion of:
Ordered Collection:

When a Collection is ordered, it means you can iterate in the collection in a specific (not-random) order (a Hashtable is not ordered).

A Collection with a natural order is not just ordered, but sorted. Defining a natural order can be difficult! (as in natural String order).

Another difference, pointed out by HaveAGuess in the comments:

• Comparable is in the implementation and not visible from the interface, so when you sort you don't really know what is going to happen.
• Comparator gives you reassurance that the ordering will be well defined.