Roughly speaking, it's about "when" you take square roots and what that means for the product you are trading. Here is a handy guide on a zoo of vol/var swap/forward/future products.

The key thing is less about what "volatility" and "variance" have been. (Realized volatility is the square-root of realised variance). We're talking about the expectation for the next month's volatility or variance.

The "mathematician" way to think about this (although I think this is a little unhelpful) is . If "X" is (future) realised variance (as yet unknown), then the former is "volatility" and the latter is "square root of variance" (what I call "variance in vol units"). Therefore "expected volatility" is lower than "square root expected variance". The difference is what needs compensating

The more practical way to think about this, is that variance is being dominated much more by the tails (or volatility of volatility). When you trade a variance, you need a premium over volatility to compensate you for these tails (even if they don't realise very often).

Another way to think about this, is there is "convexity" in variance (when measured in units of volatility). If you are long and volatility goes up, you much more (because it's squared), but if it goes down, you aren't making as much less.