A volatility swap is an agreement to exchange the realized volatility of an asset between time 0 and time T for a prespecified fixed volatility. Then realized volatility is usually calculated with the assumption that the mean daily return is zero. Suppose that there are n daily observations on the asset price during that period between time 0 and time T.
The variance swap is an agreement to exchange the realized variance rate V (bar) between time 0 and time T for a prespecified variance rate. The variance rate is the square of the volatility. Variance swaps are easier to value than volatility swaps. This is because the variance rate between time 0 and time T can be replicated using a portfolio of put and call options.