Why is the distinction between linear and context-free grammars useful?
Let $\cal F$ and $\cal G$ be two classes of grammars, or what ever language defining devices you want. We assume that $L({\cal F}) \subset L({\cal G})$, the languages defined by the first class are included in those of the second.

Why consider both ${\cal F}$ and $\cal G$?

The most obvious reason is that the "models" in $\cal F$ are usually easier to handle to those in $\cal G$. That means it is in general easier to understand, write and analyse grammars in $\cal F$ than those in $\cal G$.  So whenever it is possible to work in the smaller class we might work there.

Sometimes we want to study and understand what makes a certain type of grammar hard to handle. We know that it is undecidable whether two context-free grammars generate the same word. For regular languages (given by right-linear grammars or finite automata) it is simple to construct the intersection and decide its emptiness. Now it is a matter of curiosity to know what happens in between: what about emptyness of intersection of linear languages?

Things are not always that straightforward. Deterministic pushdown automata are strictly less powerful than the general family (which generates context-free languages). They even allow a decidable equality test which is interesting for modellers. On the other hand, except for closure under complement, they have little closure properties.