The actual difference between Charger and Adapter is, Charger is an electrical or electronic device which is specially designed to charge a chargeable device such as a battery or a super-capacitor whereas the Adapter is an electrical or electronic device which is specially designed to provide a power supply to an electronic device such as laptop or music player or USB fan. I know you did not understand, what I am trying to say. Please read the full article to understand the actual difference.
Charger
Suppose we have a battery rating of 4.2V, 3000mAh. Now we want to charge it. So to charge this battery we need a 5V DC power supply but we have 230V AC(India) or 120V AC(US) supply in our home. In this case, the charger is required which will convert the AC to DC and step down the voltage from the standard voltage to the required voltage. So the charger is specially designed to charge the battery.
Adapter
Suppose, we have a 5V DC fan. If we want to use this fan we need a 5V power supply. Here the Adapter is required. So the adapter is designed for only to provide the required power supply to the fan. If the fan was 5V AC, then we need an adapter which can give 5V AC supply as output.
Charger VS Adapter
1. The charger is specially designed to charge a device such as a battery or a supercapacitor whereas The adapter is a device which is designed to provide required power supply to a device.
2. There is no AC to AC charger available because of all the batteries and super-capacitors are charged with DC, not AC but the both AC to DC and AC to AC adapters are available in the market, they are used to provide power supply to DC and AC devices respectively.
3. Generally, the charger output voltage rating greater than the required voltage of the devices. For example, a 4.2V battery needs a 5V charger. On the other hand, adapters have the same output voltage rating as required of devices. For example, a 5V DC fan required 5V DC adapter.
Mobile Charger VS Laptop Adapter
After reading the above explanation, you may think why different names are used such as Mobile charger, laptop adapter as they are both used for charging purpose.
If you remove the battery from your mobile and connect the charger and try to ON the mobile, it will not ON because the mobile charger only design to charge the battery not to provide the power supply to the mobile for its operation. Mobile phone always takes the power supply from the battery, even the charger is connected or not.
If you remove the battery from your laptop and connect the adapter and then try to ON the laptop, it will be ON because the adapter is designed to provide the power supply to the laptop for its operation. When you connect the adapter it will not directly be connected to the battery. An additional circuit is installed inside the laptop to charge the battery taking the power supply from the adapter.
So when we connect the adapter, the laptop takes the power supply from the adapter and operate. At the same time, the battery also charges. When we disconnect the adapter, the laptop takes the power supply from the battery and operate.