Part of any issue looking at SQL vs NoSQL currently is that SQL has continued to evolve and actually taken steps to incorporate no-sql like paradigms.

A good example is JSON support. Initially if you wanted to store or manage JSON objects it was either as text in SQL or required a NoSQL database. Now the SQL standard has support for JSON.

Similarly "Big Data" is a space for NoSQL, things like columnar databases were designed for more efficient storing/processing (although columnar indexes can now exist in SQL databases I believe).

Some spaces where NoSQL still is really important is things like graph databases and key value (as others have mentioned). Graph databases require a different query language and backend.

NoSQL is best used as a key-value storage, where the value can be non-tabular or mixed data. As an example, imaging you have a session cookie value identifying a user. That user might have many different groups, roles, claims, etc. If you wanted to store that data in a RDBMS you would likely need a table for every 1-to-many data point (Session -> SessionRole, Session -> SessionGroup, etc). In NoSQL this would be represented as a single key with a json object that could looks quite different from other Session json objects. If you then need to delete that session it's a single key delete, where in the RDBMS you would have to make sure that delete chained to the downstream tables.

This type of key-value lookups are often very fast and used as a caching layer for complex data calculations as well.

The big downside to this is indexing and querying the data not by the primary key. It would be hard to find all users in a specific group as you would need to scan each key-value. It looks like NoSQL has some indexing capabilities now but when I first used it it did not.

A place where this type of DB really shines is in messaging. For example Discord uses NoSQL. Each message someone sends is a row, but each message can have reactions made on it by other users. In a SQL database there would be 2 tables, one for messages and one for reactions with a foreign key to the message. But at the scale of Discord they can't use a single SQL server which means you can't really have 2 tables and do a join to find reactions on a message. Obviously you could shard the databases. But in NoSQL you just lookup the message and the reactions are just stored alongside it, not in another table, making the problem simpler.

I spent 30 years working with derivatives of the Pick Operating System and its integrated DBMS. Notably Universe and Ultimate. Back in the day, it was very, very difficult to even explain how they worked to others because the idea of key/value wasn't commonly understood, at least as it is today.

I was surprised at how similar MongoDB is to Pick in many many respects. Basically, key/value with variant record structures. MongoDB uses something very close to JSON, while Pick uses variable length delimited records. In either case, access to a particular record in near instantaneous give the record key, regardless of how large the file is. Back in the 1980's and earlier, this was a huge advantage over most of the RDBMS systems available, as storage was much slower than today. We could implement a system that would otherwise take a huge IBM mainframe, on hardware that cost 1/10 the price.

From a programming perspective, everything revolves around acquiring and managing keys. Even index files, if you had them (and in the early days we didn't so we maintained our own cross-reference files) were just files keyed on some value from inside records from the main data file. Each record in an index file was just a list of record keys to the main data file.

Yes, you can (and we did) nest data that would be multiple tables in an SQL database into a single record. This was something called "Associated Multivalues". Alternatively, you could store a list of keys to a second file in a single field in the first file. We did both.

One thing that became very time/disk/cpu expensive was traversing an entire file. 99% of the time we were able to architect our systems so that this never happened in day to day processing.

A lot of stuff we did would horrify programmers used to SQL, but it was just a very different paradigm. Back in a time when storage and computing power were limited and expensive, the systems we built stored otherwise unthinkable amounts of data and accessed it with lightening speed on cheap hardware.

To this day, the SQL concepts of joins and normalization just seems like a huge waste of space and power to me.

This isn't a sophisticated opinion or anything, but personally I find RDBMS to be a bad fit for how data is typically structured in your program. You will usually have an object, often with sub-objects all built up like a tree. If you want to load that into an SQL DB, you need to split it up, equip lots of its parts with IDs and then hope that you can reconstruct it when you take it back out.

On the other hand, JSON was directly designed for serializing programming objects. The chance of you being able to persist and load your object with hardly any structural changes is high.

Of course, this does have other downsides, like the data not being as flexible to access. Similarily, data in an RDBMS is very structured, whereas in many NoSQL databases, you can have individual entries with different fields than the rest.
So, that's perhaps a more general takeaway: SQL makes it hard to put something into the database, but easy to get it out. NoSQL often reverses this.