The advantages and disadvantages if high ceilings differ based on climate. In a local where your concern is cooling, like Florida or California, rather than heating, like Wisconsin or Oregon the things you must take into account vary widely.
If you have high ceilings in southern or central California, where the climate is rather temperate, you can make high ceilings work to your advantage using passive thermal controls like house and attic vents and a good layer of insulation in the ceiling… in that respect although the sun would heat the air towards the top of the room, if built correctly the heat rising and leaving in the top of the room could actually pull the cooler air in through the windows. If you live in Florida, where it is almost always hot and humid, although the same issues would mitigate cooling expenses, you will not be comfortable without AC, which means you will have more space to cool. Even though there is more space to cool, the heat will tend to be towards the top of the ceiling, which will make the inhabitants more comfortable.
Heating is a totally different story, since heat rises; it may be more difficult to keep the areas with high ceilings comfortable where the occupants will actually be located. This issue can be mitigated using ceiling fans and slab heating, but depending on construction methods, may still be quite expensive.
Basic maintenance, like changing a ceiling van, replacing a light bulb can become an intensive task depending on exactly how high the ceilings are.
Beyond the heating, cooling, and additional cost of construction, there are some advantages to high ceilings.
If you live in a place where you may find yourself spending an excessive amount of time inside, you may feel less cooped up in a house with high ceilings.
In resale situations, high ceilings are oft touted as a boon for prospective house sales, and many buyers tend to be interested because of the spacious feeling typically associated with high ceilings.
If the decision is not financially based - go with what is aesthetically appealing to you. If it long term cost is your concern, discuss this with your contractor… they should be familiar with the best ways to mitigate heating and cooling expenses in your town!