Redis is a data structure server that can be used as a database server on its own, or paired with a relational database like MySQL to speed things up. Redis can be configured as a cache for WordPress to alleviate the redundant and time-consuming database queries used to render a WordPress page. The result is a WordPress site which is much faster, uses less database resources, and provides a 'tunable' persistent cache.
Nginx FastCGI module has an Nginx directive for caching dynamic content that are served through PHP backend. When a web page is cached, repeated requests for the same page by web clients is quickly returned by the webserver because the page is coming from a cached location.
Instead of the webserver compiling all the dynamic data that makes up the page, before returning it to the web clients on each request, a page is cached as a whole, and then the webserver doesn’t have to fetch dynamic data before returning that page to web clients.
So, I think I understand it. The primary difference between FastCGI and Redis is in what gets cached. Using Redis with WordPress will cache the results of common database queries and speed the display of a page. The FastCGI cache instead caches the whole page after it's been generated. Neither is necessarily better than the other as it's more a matter of what works best for your particular use case, right?