The main reason not to use an SQL database is scalability. The transactional guarantees and the relational model make it almost impossible to scale a database usefully across more than a few machines, especially given the write-heavy workloads generated by modern web applications.
An app like Facebook can't be made to work on a straightforward SQL database, except by massive partitioning and sharding, which requires significant adjustments to the app logic as well. That's why Facebook developed Cassandra.
NoSQL basically means you make do without some SQL-typical features like immediate consistency or easy joins, in exchange for being able to use a database that scales much better.
Conversely, there is no point in using NoSQL if your website never has more than a dozen concurrent users (which is true for the vast majority of all sites).
