Project Euler #8, I don't understand where I'm going wrong

In fact your solution is too small rather than too big. The answer is what was pointed out in the comments, that there is integer overflow, and the clue is in the fact that your solution is close to the largest possible value for an signed int: 2147483647. You need to use a different type to store the product.

Note that the answer below is still 'correct' in that your code does do this wrong, but it's not what is causing the wrong value. Try taking your (working) code to http://codereview.stackexchange.com if you would like the folks there to tell you what you could improve in your approach and your coding style.

Previous answer

You are checking for a new greatest product inside the inner loop instead of outside. This means that your maximum includes all strings of less or equal ton 13 digits, rather than only exactly 13.

This could make a difference if you are finding a string which has fewer than 13 digits which have a large product, but a 0 at either end. You shouldn't count this as the largest, but your code does. (I haven't checked if this does actually happen.)

for (int i=0; i < num.length() -12; i++)
{
    product = ((int) num[i] - 48);
    for (int j=i+1; j<i+13; j++)
    {
        product = product * ((int) num[j] - 48);
    }
    if (greatestProduct <= product)
    {
        greatestProduct = product;
    }
}

9^13 ≈ 2.54e12 (maximal possible value, needs 42 bit to be represented exactly), which doesn't fit into signed int. You should use int64.

If you don't want to mess with BigNum libraries, you could just take logarithms of your digits (rejecting 0) and add them up. It amounts to the same comparison.