Which one is better to boil water, using high heat or less heat? Consider that I have a water boiling kettle as below. Assume that it is made of metal, not plastic (but plastic probably also will do). Assuming I can adjust the power it uses to boil the water with the option: full or half. The question is which one is better to boil the water: using maximum power or using only half? Better here is, the water is truly and uniformly boiled, regardless of the time it consume to get the water boiled. In this case, we boil the same amount of water.
 
 A: Due to heat loss to its surroundings, it's most efficient to heat the water for as short a time as possible - this means using the maximum heat setting.
In the extreme, imagine trying to make the water reach boiling point using just a 1 watt heating element - it would never reach boiling point at all because heat losses to the surroundings would be too great.
In this case (albeit artificial) the choice is to use 3kW for a few seconds, or 1W forever. So it's clearly more efficient to use the higher heat input.
A: Your question seems to be prompted by your experience that your kettle is apparently coming to the boil, but that the coffee you then make with it is rather disappointing, as if the water wasn't hot enough. You are wondering if it is something to do with the way your kettle works...
Would you mind telling us your location? I'm not too interested in the latitude or longitude but I am rather interested in your altitude.
The boiling point of a liquid is strongly affected by air pressure and so at high altitude, you have low pressure and subsequently, a low boiling point. For example at about 3000m (10,000ft) water boils at about 90°C.
As an aside, the English mountaineer, Chris Bonnington, used to complain that he could never get decent cup of tea above Everest Base camp.
A: As Oscar said, the temperature of the boiling water dependes on the altitude of your location.
The temperature needed in your location give us, mathematically, the amount of energy that we need to boil the water, considering the dissipation, caused by the contact to recipient and atmosphere. So both, "less heat" or "more heat", can boil the water appropriately but the the time that it takes to "deliver" the enough amount of energy, that this process requires, will be different because of the power of your heater.
I think you didn't mean, by the word "better", the uniformity of the heat distribution along the water, because both will boil the water uniformly but each will take a different amount of time. Anyway, in this case using a microwave would be more appropriate. 
A: To reach a boil quickly, use a high heat.  Once you reach a boil, turn it down; you are just generating steam at a fixed temperature.
