How many of the integers between 1 and 1000, inclusive, can be expressed as the difference of the squares of two nonnegative integers?
1. **Understanding the problem**: We need to determine how many integers between 1 and 1000 can be expressed as the difference of squares of two nonnegative integers. 

2. **Expressing the difference of squares**: The difference of squares of two integers \(a\) and \(b\) can be written as:
   \[
   a^2 - b^2 = (a+b)(a-b)
   \]
   Here, \(a\) and \(b\) are nonnegative integers, and \(a \geq b\).

3. **Analyzing the factors**: For \(a^2 - b^2\) to be an integer, \((a+b)\) and \((a-b)\) must be integers. Let \(x = a+b\) and \(y = a-b\). Then:
   \[
   a = \frac{x+y}{2} \quad \text{and} \quad b = \frac{x-y}{2}
   \]
   For \(a\) and \(b\) to be integers, both \(x\) and \(y\) must be either both even or both odd.

4. **Odd numbers**: If \(x\) and \(y\) are both odd, then their product \((a+b)(a-b)\) is odd. Therefore, any odd number can be expressed as the difference of squares. The number of odd integers between 1 and 1000 is:
   \[
   \left\lfloor \frac{1000}{2} \right\rfloor = 500
   \]

5. **Even numbers**: If \(x\) and \(y\) are both even, then their product \((a+b)(a-b)\) is divisible by 4. Therefore, any even number that is divisible by 4 can be expressed as the difference of squares. The number of integers between 1 and 1000 that are divisible by 4 is:
   \[
   \left\lfloor \frac{1000}{4} \right\rfloor = 250
   \]

6. **Summing the counts**: The total number of integers between 1 and 1000 that can be expressed as the difference of squares is the sum of the number of odd integers and the number of even integers divisible by 4:
   \[
   500 + 250 = 750
   \]

Thus, the total number of integers between 1 and 1000 that can be expressed as the difference of squares of two nonnegative integers is \(\boxed{750}\).