The basic energy requirements of homes, industries etc are measured in watts. It may be sometimes seen how much energy each sector utilizes over a span of time, but generally we see how many watts are used in homes, industries then further in colonies, cities etc.
Atleast here in India (and I think it would be similar everywhere) we get our home supplies based on a base wattage that we chose; Since we also get very regular powercuts in most part of the cities nearly all homes are equipped by gensets, even these gensets are categorised by their wattage.
I even saw an old documentary yesterday about energy crisis and renewable energy sources, and while talking about any source they talked about energy generation in terms of power.
Although the total energy stored is an important aspect just think that you have a huge battery that can store large amount of energy but has very little power, then you can not use it in your home or city or anything for that matter. However if the power is equal or more than what you need, only then you can operate your appliances with ease may that be for a small or long time.
So in accordance with all above I find that measuring power is a significant method to judge a battery because power would be deciding whether it can be used or not. The total energy may define how long it can be used but power would define if we can use it at all!