Should an aspiring developer 'specialize' in a certain area, or in a certain language? Is there more of a benefit to knowing a bit of everything?
It all depends on the firm you are working for or on the product you are working on. Each firm have their own way of developing things. For example, some old product based companies are still working on languages like C++ and Java while the startups are experimenting with a whole new (read unconventional) set of languages like Ruby on Rails or even Scala.
Now it all boils down to your interest. If you love coding in Java, you will certainly find firms who will hire you solely based on your knowledge of one language (Java). But it can become a bottleneck if you ever try to move to a different field/firm. So, little knowledge of all the languages won't hurt.
The core concepts in almost all the languages is similar. You usually just need to familiarize yourself with different syntax.
The answer to your question is yes. While working in a firm on a particular product, it is highly likely that you just focus on one area. For example, I work on Java in the back-end and JS, CSS, etc. on the front-end. That's because my product is built using these languages and there is no scope of working in any other language.