For continued impractical applications, 2 digit squares allow you to leverage.
ab^2 = a^2 + 2ab + b^2
which gets you to 4 digit squares in a very tidy way.
3456^2 &= 34^2 + 2*34*56 + 56^2\\
&= 1156|0000| + 3808 |00| + 3136\\
Similarly as you said difference of squares give you a fast path to multiplication using your well known facts
55 * 59 &= 57^2 - 4 \\
Between the two you have a strong set of additional tools for multiplication of larger amounts pairing up digits and reducing the number of actual calculations to half or better in some cases.
From a practical sense, I suppose it allows you to know how many square feet of paint you need quickly I suppose.
From a third spot, I find that thinking about numbers in different ways develops “practiced” numeracy. It doesn’t seem to stick around if you don’t practice but while you are practicing you can get a bit of a taste for how smart people play with abstractions.
There is a similar kind of feeling to the Soroban.
If you could, but I can’t, visualize your calculations effectively while having an intuitive grasp of algebra it might be very entertaining. I wouldn’t compare it to playing the piano but it would be nice to be adept with numbers and digging into learning squares, exponents, primes, logarithms, integration, algebra as mental exercise is a pleasant hobby. Squares are kind of a minor gateway drug.