What is the importance of linear languages? What is the point of linear languages? They appear to be an intermediate set of languages in between regular and context-free languages, but do they have any useful or nice properties that either have been studied, or make them worthwhile to study?
The first obvious reason why linear languages were introduced is that mathematicians can hardly resist, facing a lateralized notion, to consider the symmetrical version as well. For instance, in algebra, you deal with left or right inverses, but also with inverses. In ring theory, you consider left or right ideals, but also ideals, etc.

Since left linear and right linear grammars define regular languages, studying linear grammars just came as a natural question to study in the early days of formal language theory. Furthermore, as you mentioned in your question, linear languages appeared to be a natural intermediate class between regular and context-free languages.

A nice characterization of linear languages was given by Arnold L. Rosenberg: A machine realization of the linear context-free languages, Information and Control 10 (1967) 175–188. The formulation below is Proposition V.6.5 in J. Berstel's book, Transduction and Context-free Languages, Teubner 1979.

A language $L$ of $A^*$ is linear if and only if there exists a rational subset $R$ of $A^* \times A^*$ such that 
$$
L = \{ u\tilde v \mid (u, v) \in R \}
$$
where $\tilde v$ denotes the reversal of $v$. This result shows that, in a very loose sense, linear languages are a generalization of the language of palindromes.