Databases that predate Codd's work, or long-established ones like dbm and BDB, may appear similar to modern-day NoSQL databases in how they operate, but they surely aren't the same.
Those systems couldn't use relational theory or SQL because they didn't exist yet, or at least didn't reject them outright as one of their main goals.
NoSQL databases, on the other hand, are completely about rejecting relational theory and rejecting SQL. That's at the very core of their philosophy.
True NoSQL databases have been developed as a reaction to several things:
1. A very, very, very small number of situations where relational DB systems cannot easily scale.
2. The far more widespread ignorance of the basics of relational theory, and a lack of willingness to learn about it.
3. The far more widespread ignorance of SQL, and a lack of willingness to learn about it.
4. An urge to be "different" solely for the sake of being different, even if it brings no technological benefit.
5. Unmitigated hype surrounding the term "NoSQL".
Of those, 1) is perhaps the only legitimate reason for using a NoSQL approach today. The number of times this sort of a situation actually arises is remarkably small.
The other four are why those databases have become more widely used, especially within the web development community. As anyone who has dealt with such systems knows, they're rarely about safely and reliably storing and managing data, and they're rarely about doing this efficiently. They're merely a shortcut that some developers use to avoid learning how to use a RDMS.