As you identified, no-cache doesn't mean there is never caching, but rather that the user agent has to always ask the server if it's OK to use what it cached. By contrast, no-store says to not even keep a copy, which means there's nothing to ask about. If you know the answer to "Can I reuse this?" is always no, you get a performance boost by skipping cache validation and saving room in the cache for other data.
Aside from performance, there is a behavior difference with browser history. HTTP 1.1 section 13.13 says that "expiration time does not apply to history mechanisms." The no-cache header describes expiration, and so doesn't apply to history mechanisms such as the back button. Thus, the user can navigate backward to a previous page with no-cache without the server being contacted.
The no-store header, on the other hand, prevents the data from being stored outside of a session, in which case it simply isn't available for a history mechanism to use. With no-store, if the user ends his session by navigating to another domain and then goes back, the only way for browser to know what to display is to get the initial page again from the server.
Here's how a Chromium issue on this topic makes the distinction:
no-cache doesn't mean "don't cache this" (that would be no-store). no-cache means don't use this for normal loads unless the resource is revalidated for freshness. History navigations are not normal loads.