What are the differences between a Java enum and a class with private constructor?

I was trying to understand how Java enum really works and I have come to the conclusion that it is very similar to a normal Java class that has its constructor declared private.

I have just come to this conclusion and it isn't based on much thinking, but Id like to know whether I miss anything.

So below is an implementation of a simple Java enum and an equivalent Java class.

public enum Direction {
    ENUM_UP(0, -1),
    ENUM_DOWN(0, 1),
    ENUM_RIGHT(1, 0),
    ENUM_LEFT(-1, 0);


    private int x;
    private int y;

    private Direction(int x, int y){
        this.x = x;
        this.y = y;
    }
    public int getEnumX(){
        return x;
    }
    public int getEnumY(){
        return y;
    }
}

What is the difference in meaning between the code above and below?

public class Direction{
    public static final Direction UP = new Direction(0, -1) ;
    public static final Direction DOWN = new Direction(0, 1) ;
    public static final Direction LEFT = new Direction(-1, 0) ;
    public static final Direction RIGHT = new Direction(1, 0) ;


    private int x ;
    private int y ;

    private Direction(int x, int y){
        this.x = x ;
        this.y = y ;
    }
    public int getX(){
        return x;
    }
    public int getY(){
        return y;
    }
}