One (stddev) is in the same units as the quantity being measured, while the other isn't. I know nothing about machine learning and suck at math and statistics. Did I pass?
But anyway, you'd get enough for a follow up question, but not really enough to pass. The key is that the sum of uncorrelated distributions has a variance that is the sum of the variance of those distributions. This isn't true for standard deviation. This is useful so we use variance when we think we're dealing with random variables. Standard deviation isn't as useful for intermediary processes.
Another follow up question - actually the first question in my head, but by a mental slip-up I typed out the wrong thing: Why do we use square error when absolute error would do?
The answer to this is actually quite illuminating because there are multiple valid but contradictory positions on it, much like the Bayes vs Frequentist debate.
What makes you think this? Anything aside from intuition?