Average-price options versus plain European options

Compared with an otherwise identical European vanilla option, is an Asian option (whose payoff uses the average price over the option's life) typically more or less expensive?

Cheaper. An average is less volatile than the terminal price (for a continuously sampled arithmetic average, roughly $\sigma/\sqrt3$), and option value increases with the volatility of the quantity in the payoff. Averaging also smooths out end-of-life spikes, so Asian calls and puts generally cost less than vanillas.