What is the difference between CHARACTER VARYING and VARCHAR in PostgreSQL?

The short answer: there is no difference.

The long answer: CHARACTER VARYING is the official type name from the ANSI SQL standard, which all compliant databases are required to support. (SQL compliance Feature ID E021-02.)

VARCHAR is a shorter alias which all modern databases also support. I prefer VARCHAR because it's shorter and because the longer name feels pedantic. However, postgres tools like pg_dump and \d will output character varying.

VARCHAR is an alias for CHARACTER VARYING, so no difference, see documentation :)

The notations varchar(n) and char(n) are aliases for character varying(n) and character(n), respectively. character without length specifier is equivalent to character(1). If character varying is used without length specifier, the type accepts strings of any size. The latter is a PostgreSQL extension.

Note on capitalization: The PostgreSQL documentation uses the all lower case stylization: character varying. In contrast the official SQL standard uses the stylization with all caps throughout its 1000 pages: CHARACTER VARYING.

The PostgreSQL documentation on Character Types is a good reference for this. They are two different names for the same type.