Adding more application servers works for a while, but the effect of adding more application servers is that you increase contention on the database server.
Database servers are much harder to scale by throwing more computers at the problem. The ACID properties that underpin how databases are used do not scale across many servers. Think about simultaneously applying the results of a transaction across 10 servers, and then what happens when there are 100,000 simultaneous transactions in progress.
That is where distributed databases step in. These products do not do ACID, they do CAP. Effectively you have to choose two characteristics from consistency, availability and partition tolerance and allow the third one to slip.
The programming model is usually eventual consistency, so changes get distributed between the database servers over time. (usually in ms timescales), however this is not appropriate everywhere.
Some systems claim to offer all three CAP properties, but I have yet to see one that truely achieves this
The distributed database does not do SQL, so the data access layer has tone rewritten. This kind of change takes time to implement, and requires new ways of thinking.