c# array [][] vs [,]

One is an array of arrays, and one is a 2d array. The former can be jagged, the latter is uniform.

That is, a double[][] can validly be:
double[][] x = new double[5][];

x[0] = new double[10];
x[1] = new double[5];
x[2] = new double[3];
x[3] = new double[100];
x[4] = new double[1];

Because each entry in the array is a reference to an array of double. With a jagged array, you can do an assignment to an array like you want in your second example:
x[0] = new double[13];

On the second item, because it is a uniform 2d array, you can't assign a 1d array to a row or column, because you must index both the row and column, which gets you down to a single double:
double[,] ServicePoint = new double[10,9];

ServicePoint[0]... // <-- meaningless, a 2d array can't use just one index.

UPDATE:

To clarify based on your question, the reason your #1 had a syntax error is because you had this:
double[][] ServicePoint = new double[10][9];

And you can't specify the second index at the time of construction. The key is that ServicePoint is not a 2d array, but an 1d array (of arrays) and thus since you are creating a 1d array (of arrays), you specify only one index:
double[][] ServicePoint = new double[10][];

Then, when you create each item in the array, each of those are also arrays, so then you can specify their dimensions (which can be different, hence the term jagged array):
ServicePoint[0] = new double[13];
ServicePoint[1] = new double[20];

Hope that helps!