Reviewing Facade in the GoF book and the link in another answer to Martin Fowler’s Gateway, it appears that their focus is in opposite directions.
Facade provides a simple uniform view of complex internals to (one or more) external clients;
Gateway provides a simple uniform view of external resources to the internals of an application.
This distinction lets us focus on which is more important in a design:
With the Facade, the external system is our customer; it is better to add complexity facing inwards if it makes the external interface simpler.
With the Gateway, the internal system is our customer; give it all the help we can, even if the externals are more complex.