Why use getters and setters?

• Encapsulation of behavior associated with getting or setting the property - this allows additional functionality (like validation) to be added more easily later.
• Hiding the internal representation of the property while exposing a property using an alternative representation.
• Insulating your public interface from change - allowing the public interface to remain constant while the implementation changes without affecting existing consumers.
• Controlling the lifetime and memory management (disposal) semantics of the property - particularly important in non-managed memory environments (like C++ or Objective-C).
• Providing a debugging interception point for when a property changes at runtime - debugging when and where a property changed to a particular value can be quite difficult without this in some languages.
• Improved interoperability with libraries that are designed to operate against property getter/setters - Mocking, Serialization, and WPF come to mind.
• Allowing inheritors to change the semantics of how the property behaves and is exposed by overriding the getter/setter methods.
• Allowing the getter/setter to be passed around as lambda expressions rather than values.
• Getters and setters can allow different access levels - for example the get may be public, but the set could be protected.

For example, if foo was public, I could easily set it to null and then someone else could try to call a method on the object. But it's not there anymore! With a setFoo method, I could ensure that foo was never set to null.

Accessors and mutators also allow for encapsulation - if you aren't supposed to see the value once its set (perhaps it's set in the constructor and then used by methods, but never supposed to be changed), it will never been seen by anyone. But if you can allow other classes to see or change it, you can provide the proper accessor and/or mutator.