The heat pump will heat and cool your house. The air condition unit will only cool your house. A heat pump, does just want is says, it will pump heat our of your house or heat into your house. Not a traditional pump like a water pump but removes heat or adds heat to the area.

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A heat pump is a popular style of air conditioner which has the capability to heat with the refrigeration cycle as well as to cool with it. The biggest differences are the refrigerant reversing valve, a defrost feature and the dual -action room thermostat used in and with the A/C heat pump. The functions of the evaporator and the condenser are swapped so that heat is removed from the outside air and released through the inside coil..
Depending on the local temperature, a heat pump will heat without the need of supplemental heating be it electrical resistance heaters built into the unit or auxiliary gas or oil-fired heaters. Its coeficient of performance will be around 3 to 1 when heat is begun to be needed , but will diminish with colder weather. I would not recommended anything but an available heat pump in moderate climates because of the simplicity of the installation and the low energy heating capability for the many cold days during which there are few temperatures below about 25F. Even then, the heat pump will be performing well until about 15 F. Supplemental heating (electric, gas, oil) will be needed , but the heat pump will have delayed that need and done so with a power saving in the meanwhile. The same might be said even for colder climates because most of the time the low temperatures are still occurring a low percentage of the time.
If gas is a less expensive heating medium than electricity, then a combination of systems could be recommended.
If gas is really less expensive than electric power for the whole year, perhaps more of the gas -heating / gas- fired absorption coolers will be made available.

Some people do not like a heat pump because they have been used to hot air blowing out the grilles while warm air is discharged by the heat pump.

The heat pump uses a reverse air conditioning cycle to deliver heat. The refrigerant, or freon as you may call it, is compressed, and sent to the inside coils where it gives up it’s heat to the inside air, and returns to the outside coils where, hopefully, the refrigerant will become a gas a again, to be re-compressed, and returned to the inside coils. If the refrigerant does not become a gas soon enough, it can damage, or destroy the compressor. To this end, extra heating coils are used to prevent this, making the heat pump, in colder climates, less efficient. An air conditioning unit is just an air conditioner. It compresses the refrigerant, which goes through the outside coils to become a liquid, goes inside to the indoor coils where it picks up heat by turning into a gas again, and returns to the compressor to begin the cycle again. Your best choice, prices of fuel not with standing, is to use a gas furnace, and a regular air conditioner for your central air conditioning system. Gas, is a better heating fuel, than a heat pump, or resistance wire heating system.