What is the difference between the Facade and Adapter Pattern?

The Facade Pattern wiki page has a brief note about this.

"An Adapter is used when the wrapper must respect a particular interface and must support a polymorphic behavior. On the other hand, a facade is used when one wants an easier or simpler interface to work with."

I heard an analogy that you should think of your universal remote control that you've set up to work with all your different stereo systems - you press "on" and it turns on your cable box, your receiver, and your TV. Maybe it's a really fancy home theater and it dims the lights and draws the shades too. That's a Facade - one button/function that takes care of a more complicated set of steps.

The Adapter pattern just links two incompatible interfaces.

EDIT: A quick analogy for the Adapter pattern (based on the comments) might be something like a DVI-to-VGA adapter. Modern video cards are often DVI, but you've got an old VGA monitor. With an adapter that plugs into your video card's expected DVI input, and has its own VGA input, you'll be able to get your old monitor working with your new video card.

Adapter == making a square peg fit into a round hole.

Facade == a single control panel to run all the internal components.

Facade:

Key takeaways : ( from journaldev article by Pankaj Kumar)

1. Facade pattern is more like a helper for client applications
2. Facade pattern can be applied at any point of development, usually when the number of interfaces grow and system gets complex.
3. Subsystem interfaces are not aware of Facade and they shouldn’t have any reference of the Facade interface
4. Facade pattern should be applied for similar kind of interfaces, its purpose is to provide a single interface rather than multiple interfaces that does the similar kind of jobs

Facade class diagram:

Adapter:

1. It is a structural pattern
2. It is useful to work with two incompatible interfaces
3. It makes things work after they're designed

Class diagram of Adapter:

You can find more details about Adapter in this SE post:

Difference between Bridge pattern and Adapter pattern

Key differences:

1. Facade defines a new interface, whereas Adapter uses an old interface. Adapter makes two existing interfaces work together as opposed to defining an entirely new one
2. Adapter and Facade are both wrappers; but they are different kinds of wrappers. The intent of Facade is to produce a simpler interface, and the intent of Adapter is to design to an existing interface

Have a look at sourcemaking article too for better understanding.

The purpose of a

facade is simplicity

adapter is interoperability.