For instance, graphs are totally doable with a traditional rdbms, however it is painful. You end up joining a table against itself (or via an edges table) multiple times, or alternately bouncing many queries off the table as you iterate the graph. One common type of NoSQL db is the graph database that is designed with graphs in mind and you don't even have to think about this access. It is nice.
Another case that you can do with traditional RDBMS but is annoying is loose user defined fields, such as "tags" where you have to create a tags table and a join table to make it work, with a lot of potential inefficiency there (even with indexes). Or even worse, when you have user defined attributes - lots of custom table creation per user, or big joins against a star topology to do it properly. (Or if not, you end up with something that looks like an sql database to a bunch of frontend code).
Of course other times you'll find yourself doing something in a NoSQL database that is effectively doing a bunch lookups against a table then combining a result (at which point switching to RDBMS is the solution)...
I guess what I'm saying is that lately the data store is being looked at as a component more like a library than a subsystem. I'm not sure if this is good or bad, but it certainly has helped a few cases I've dealt with nicely - ripping out a horribly complex data layer and replacing it with a NoSQL solution where the data model is shaped like my data.
Basically it's a matter of the right tool for the job.
Relational databases partly offer solutions for this, too. But in a relational database, these things are (often clumsy) extensions and not well supported.
It means thinking about what and how you're going to do something before you start coding (design up front). It means creating a throw-away proof of concept before you start coding. But it is flexible, extensible, and changes to the schema, as it were, do not impact up-stream dependencies.
The term "NoSQL" database is a bit problematic, because the definition only says that the database is not relational, but the average NoSQL database has other differences with the average SQL database: using JSON and JavaScript, and perhaps queries with HTTP and so on.
The point is not "what is possible", but choosing the best tool for the job.
SQL Express runs as a single instance on a client so you get all three - consistency, availability and partition tolerance[1]. When running something else on more than one node you can choose two of those three. Availability and consistency are usually chosen because of business drivers. If partition tolerance is required (I've yet to encounter a scenario that makes a compelling case for it [2]) then eventual consistency is the price.
There are databases out there that will suite any combination of the three. Unfortunately the equation is somewhat more complex because databases that offer consistency and partition tolerance (for example) don't typically offer JOIN -like functionality.
Everything's a trade-off. Other considerations are tried and tested v. bleeding edge; painful v. painless; ideal v. affordable; and so forth.
[1] http://en.wikipedia.org/wiki/CAP_theorem
[2] I'm not saying there aren't scenarios where it makes sense (Google, for example) - just that I've not encountered one. I do run into many people that want Mongo and, not understanding what they're asking for would be better of with a relational database.
Indexes can be created on fields within JSON documents for faster access.
Online ALTER TABLE doesn't take any long running lock, so it can run in the background without affecting your queries.
That doesn't mean one is better than the other, it's two alternative ways of achieving things.
Sometimes a document collection is just a whole lot easier to reason about.
"Wouldn't it be cool to have a multipurpose database which we would be able to query with a language, but not SQL, because SQL sucks, for some reason.".
Put it differently, what does ArangoDB, MongoDB, whateverDB bring that relational databases didn't bring 30 years ago?