To compare the number of distinct prime divisors of the two given expressions, we need to analyze each expression carefully.

1. **Expression 1: $200^2 \cdot 201^2 \cdot \ldots \cdot 900^2$**

This expression is the product of the squares of all integers from 200 to 900. The prime divisors of this product are simply the prime numbers that appear in the factorization of any number between 200 and 900.

Let's denote the set of prime numbers between 200 and 900 as $P_1$. The number of distinct prime divisors of this product is the number of elements in $P_1$.

2. **Expression 2: $(200^2 - 1)(201^2 - 1) \cdot \ldots \cdot (900^2 - 1)$**

Each term in this product can be factored using the difference of squares:
$$n^2 - 1 = (n - 1)(n + 1)$$
Therefore, the expression becomes:
$$(199 \cdot 201)(200 \cdot 202) \cdot \ldots \cdot (899 \cdot 901)$$

This product includes all integers from 199 to 901. Let's denote the set of prime numbers between 199 and 901 as $P_2$. The number of distinct prime divisors of this product is the number of elements in $P_2$.

3. **Comparison of $P_1$ and $P_2$:**

- $P_1$ includes all prime numbers between 200 and 900.
- $P_2$ includes all prime numbers between 199 and 901.

The only additional primes in $P_2$ that are not in $P_1$ are 199 and 901.

- 199 is a prime number.
- 901 is not a prime number (since $901 = 17 \times 53$).

Therefore, the set $P_2$ has one more prime number (199) than the set $P_1$.

Conclusion:
The number of distinct prime divisors of the second expression is greater by one compared to the first expression.

The final answer is the number of distinct prime divisors of the second expression is greater by one compared to the first expression.