What's the advantage of a Java enum versus a class with public static final fields?

Question

I am very familiar with C# but starting to work more in Java. I expected to learn that enums in Java were basically equivalent to those in C# but apparently this is not the case. Initially I was excited to learn that Java enums could contain multiple pieces of data which seems very advantageous (http://docs.oracle.com/javase/tutorial/java/javaOO/enum.html). However, since then I have found a lot of features missing that are trivial in C#, such as the ability to easily assign an enum element a certain value, and consequently the ability to convert an integer to an enum without a decent amount of effort (i.e. Convert integer value to matching Java Enum).

So my question is this: is there any benefit to Java enums over a class with a bunch of public static final fields? Or does it just provide more compact syntax?

EDIT: Let me be more clear. What is the benefit of Java enums over a class with a bunch of public static final fields of the same type? For example, in the planets example at the first link, what is the advantage of an enum over a class with these public constants:

public static final Planet MERCURY = new Planet(3.303e+23, 2.4397e6);
public static final Planet VENUS = new Planet(4.869e+24, 6.0518e6);
public static final Planet EARTH = new Planet(5.976e+24, 6.37814e6);
public static final Planet MARS = new Planet(6.421e+23, 3.3972e6);
public static final Planet JUPITER = new Planet(1.9e+27, 7.1492e7);
public static final Planet SATURN = new Planet(5.688e+26, 6.0268e7);
public static final Planet URANUS = new Planet(8.686e+25, 2.5559e7);
public static final Planet NEPTUNE = new Planet(1.024e+26, 2.4746e7);

As far as I can tell, casablanca's answer is the only one that satisfies this.

Answer 1

1. Type safety and value safety.
2. Guaranteed singleton.
3. Ability to define and override methods.
4. Ability to use values in switch statement case statements without qualification.
5. Built-in sequentialization of values via ordinal().
6. Serialization by name not by value, which offers a degree of future-proofing.
7. EnumSet and EnumMap classes.