Redis is the primary key store for tokens, policies, quotas and rate limit information. If you are running 10 gateway nodes and fielding hundreds of simultaneous connections from the same user, then you need to synchronise the rate limiter and quota count across instances.
Redis is very very very fast, so it's excellent for these kinds of counters.
Tyk CE is a headless API gateway, you administer it via config files and the API.
Tyk Pro adds the Tyk Dashbord and Analytics service, the developer portal and a fine-grained admin API on top of the regular gateway. It also comes bundled with our Premium support offering.
They both use Redis and Pro uses CE as the DMZ gateway, but adds a management layer so you can administer and push API configs, tokens, users etc to a cluster of a Tyk nodes from a single UI.
Hope that helps.