One of the first examples of a cluster algebra given in Fomin and Zelevinsky's original paper is the homogeneous coordinate ring $\mathbb{C}[G_{2,n}]$ of the Grassmannian of planes in $\mathbb{C}^n$. It has also been shown by Scott that $\mathbb{C}[G_{k,n}]$ carries a cluster algebra structure for any $k$.

When I'm explaining cluster algebras to somebody for the first time, I usually start with these examples, because they're the ones I spend the most time thinking about. Another reason is that the $\mathbb{C}[G_{2,n}]$ case can be nicely visualized by identifying clusters of Plücker coordinates with triangulations of the $n$-gon.

A common response to this example is "that's very pretty, but what does it tell us about $\mathbb{C}[G_{2,n}]$ as an algebra?", which is a reasonable question I don't feel I can answer well, if at all. As I'm probably going to be giving several talks about cluster algebras in the next few months, it would be nice to have a good answer to this.

Essentially all I can say so far is that the cluster monomials form a distinguished linearly independent set, and in the case of $\mathbb{C}[G_{2,n}]$ they are even a basis, but this isn't hugely satisfactory.

So, my question is:

Are there any results about $\mathbb{C}[G_{k,n}]$ proved using the cluster algebra structure that weren't known before this additional structure was discovered?

If there are better answers for different algebras then I would also be interested, but for the purposes of asking a hopefully-not-too-vague question, I'll stick to Grassmannians.