Why is case sensitivity in passwords important?

If your answer is: because 62^8 (^12, ^16) is much bigger than 36^8 (^12, ^16) then I assume you don't actually know how people crack passwords, which isn't brute-force.

If your system is capable of being brute-forced, you already failed, because there's no reason to give someone enough guesses that a brute-force approach succeeds.