Redis gives you fundamental data structures (lists, sets, hashes - plus a few other like zsets and hyperloglogs) that represent a powerful way of expressing computation. I'm not acquainted with the full spectrum of Postgres' data structures, so I cannot do a fair comparison; all I can say is that Redis is awesome and I can't get enough of it.

Postgres essentially gives you one top-level data structure, the table. For each column however it has a wild variety of data types and structures. It similarly has a large variety of indexes, making it possible to do performant lookups on all those datatypes as your use-case requires.