Use math

public static class DifferenceOfSquares
{
    public static int CalculateSquareOfSum(int max)
    {
        var sum = max * (max + 1) / 2;
        return sum * sum;
    }

    public static int CalculateSumOfSquares(int max) =>
        (max * (max + 1) * ((max * 2) + 1)) / 6;

    public static int CalculateDifferenceOfSquares(int max) =>
        CalculateSquareOfSum(max) - CalculateSumOfSquares(max);
}

Method: CalculateSquareOfSum()

The square of the sum is defined as: sum all numbers from 1 to the provided max, and then square that number.

The formula to calculate the sum of numbers 1 to max is: max * (max + 1) / 2. We can directly convert this to code and then return its square:

var sum = max * (max + 1) / 2;
return sum * sum;

Method: CalculateSumOfSquares()

The calculate the sum of the squares, we can use the following formula: (max * (max + 1) * ((max * 2) + 1)) / 6. As this has the summation and squaring built-in, we can directly return this formula in code:

public static int CalculateSumOfSquares(int max) =>
    (max * (max + 1) * ((max * 2) + 1)) / 6;

Method: CalculateDifferenceOfSquares()

The CalculateDifferenceOfSquares() method is nothing more than calling the two methods we just created:

public static int CalculateDifferenceOfSquares(int max) =>
    CalculateSquareOfSum(max) - CalculateSumOfSquares(max);