This is something that confuses a lot of people and leaves them unsure of which to get when they are looking to buy one. In reality there is no difference between them, a psychrometer is actually a type of hygrometer.
A hygrometer is a device that is used to measure the moisture content of the air, basically this means the relative humidity and the dew point. There are several different ways that this can be done. The oldest and the most common is to use a dry bulb and a wet bulb thermometer, this is what is known as a psychrometer.
In order to determine the relative humidity and dew point with a psychrometer you need to have a dry bulb thermometer and a wet bulb thermometer so that you can compare the difference. A wet bulb is simply a thermometer that has a muslin that is soaked in water over the bulb part. This works because the latent heat of evaporation will cause the wet bulb temperature to read lower than the dry bulb, this difference can then be used to find the relative humidity and dew point by looking them up in a chart.
Professional weather stations will use a psychrometer to determine the dew point and relative humidity as they are the most accurate option available. They usually do this by having the two thermometers permanently mounted in a Stevenson Screen. It is also possible to do it with a sling psychrometer which is a portable device that holds the two thermometers, when you want to measure the moisture content of the air you spin the psychrometer in order to allow the thermometers to get an accurate temperature reading.
There are other ways that you can measure the moisture content of the air, these include changes in capacitance, resistance or thermal conductivity. These are measured with electronic devices that are usually labelled as hygrometers although you will often see them listed as dew point meters and confusingly sometimes even as psychrometers. The simple answer to the difference between a hygrometer and a psychrometer is that the electronic versions are hygrometers the ones with liquid thermometers are psychrometers.
The next question then is obviously which is the best choice for measuring the moisture content of the air. This will depend on your needs. If you are a meteorologist or somebody else for whom accuracy is the most important consideration you are going to need a psychrometer. There are others who need to determine the moisture content of the air like HVAC technicians and in some cases painters will value convenience over complete accuracy. For them an electronic hygrometer is usually the best option as it is easy to read and it works quickly. For most users you don’t need accuracy to a tenth of a degree so the electronic version is fine.
The best choice if you are somebody who is simply interested in the weather is probably neither a hygrometer or a psychrometer. Most home weather stations have a hygrometer feature and given that they can measure all of the other weather parameters they make a good option for most home users.