What is the difference between a power supply and a charger?

The two devices are similar and the difference lies in the parameters that are set and which can vary. The power supply provides a voltage with a fixed and unchanging value, e.g. 12 volts, and the current that flows is determined by the load. The current can be small or large, but the voltage in the power supply does not change. In a charger, the regulation concerns the output current, which is unchanging and is, for example, 1A. The charger output voltage, on the other hand, can vary over a wide range. This voltage is determined by the attached battery, when discharged it is small, when charged it is large.

A charger and a power supply look similar, and sometimes it is unclear whether what you have in your drawer is a charger and what is a power supply. Such confusion is due to the way electronic devices are built. In general, a charger consists of two parts: the power supply and the electronic circuit responsible for charging and controlling the state of the battery (charge monitoring circuit). This control circuit can be inside the power supply or sewn into the device.
A mobile phone, for example, has such a charge monitoring circuit built-in. Therefore, it only requires a regular power supply to be connected for charging and it takes care of the rest. In this case, the phone works in conjunction with the power supply, although the colloquial term for such a power supply is "charger".
In the second case, the circuit that supervises charging can be built into the power supply. An example of such a charger is a car charger, which controls the charging process completely and requires connection directly to the battery terminals. Such a device is also colloquially referred to as a charger, which sometimes makes it difficult to get to grips with.

How do I know where the charge monitoring circuit is?
The charging supervisor circuit is where the charging indicator (symbol on the display, LED, etc.) is located. On a phone, it is displayed on the screen (battery symbol), so that is where the supervisory circuit is located. In car battery chargers, on the other hand, the charging indicator is inside the charger housing (e.g. a red LED). So then the supervisory circuit is located inside the charger housing.
In the case of power tools, sometimes the LED showing charging is in the housing of the power supply and in other cases it is mounted in the tool housing (e.g. drill). In the latter case, a normal power supply is used for charging, because the charging part is mounted inside the drill by the manufacturer.