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

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.

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.

Technically one could indeed view enums as a class with a bunch of typed constants, and this is in fact how enum constants are implemented internally. Using an enum however gives you useful methods (Enum javadoc) that you would otherwise have to implement yourself, such as Enum.valueOf.