The heat pump technology is the same of air conditioners.
It’s common to hear that heat pumps are air conditioners able to function in reverse. And that’s true, if we assume that air conditioners only produce cold while heat pumps produce heat. But since most modern "air conditioners" and "heat pumps" provide both heating and cooling, the difference becomes increasingly meaningless…
The difference – if it exists – is that air conditioners are especially designed for cooling, while heat pumps are especially designed for heating. Air source heat pumps have, typically, electric resistance elements to provide heat in sub-freezing temperatures, when their compressors are ineffective, and AC may not have these elements. Similarly, air conditioners can be especially designed to get better SEERs/EERs (cooling efficiency coefficients).
There is little or no difference between modern air heat pumps and AC, namely in mid and low-efficient models.
In other words: in some sense, heat pumps are optimized for heating while air conditioners are optimized for cooling; in practical terms there is little or no difference between modern heat pumps and AC, namely in mid and low-efficient models, where the heating and the cooling functions aren't optimized.
Air conditioners vs. Ground-Source heat pumps
ROOM VS CENTRAL
Heat pumps - like air conditioners - can be ductless (mini-split units) or central systems. Ductless heat pumps are relatively inexpensive, depending on the number of air handlers (they may heat/cool up to four zones/rooms). Central ducted units - like central AC - are a lot more expensive ($3.000-$10,000, installed). The efficiency of ducted heat pumps - like AC efficiency - depends greatly on the quality of installation and the size of the ductwork and refrigerant charge. Improperly installed heat-pumps (or central AC) will perform badly.
Even ground-source (geothermal) heat pumps are basically air conditioners in their core.
There is a large difference in terms of costs and energy-efficiency between geothermal heat pumps and the air source units, which is relevant and should not be undervalued; and there is the special coil of the geothermal units, buried in the ground or submerged in a water body… But they too use the same refrigerant compressor technology and the same physic concepts of air conditioners.
Ground heat pumps are at their core powerful central air conditioners with a long-buried loop.
The bottom line
Choosing between an air source heat pump and an air conditioner is often a question of nomenclature.
The more important is often the installation, the ducts, the size of the equipment, the airflow, or the refrigerant charge - and obviously, the quality and energy-efficiency of the equipment.
When choosing, consider a certified model (Energy Star...) and the heating and the cooling coefficients of the machine; and take into account your climate.
Choose a model with a high COP or HSPF if you live in a hot region. And consider a high SEER/EER if your cooling needs are significant. Also prefer units able to provide zoning (room-by-room heating and cooling) and with variable speed motors… See, on these issues: Heat Pumps efficiency and AC efficiency.
And do not forget that ductless models are the best choice in low-energy consumption homes, and that you should insulate and air-seal your home carefully all over its shell and pay special attention to the windows - in order to reduce your heating and cooling needs as much as possible, and install the smallest system possible.