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Solution:

Begin by reducing 240 to its prime factors.

$\begin{array}{l}\Rightarrow 240=24\times 10\\ \Rightarrow 240=6\times 4\times 2\times 5\\ \Rightarrow 240=2\times 3\times 2\times 2\times 2\times 5\\ \Rightarrow 240=2^4\times 3^1\times 5^1\end{array}$

Next, add 1 to each of the exponents and multiply the results: (4 + 1)(1 + 1)(1 + 1) = (5)(2)(2) = 20. Thus, 240 has a total of 20 factors, which means x = 20.

Next, let’s find the prime factorization of 120.

$\begin{array}{l}\Rightarrow 120=12\times 10\\ \Rightarrow 120=4\times 3\times 2\times 5\\ \Rightarrow 120=2\times 2\times 3\times 2\times 5\\ \Rightarrow 120=2^3\times 3^1\times 5^1\\ \end{array}$

After adding 1 to each exponent and multiplying the results, we get (3 + 1)(1 + 1)(1 + 1) = (4)(2)(2) = 16. Thus, 120 has 16 different factors, which means y = 16.

Finally, we need to determine x² $–$ y²:

$\Rightarrow x^2-y^2={20}^2-{16}^2=400-256=144$