Why does the von Mises-Fisher distribution need two parameters? The von Mises-Fisher distribution has two parameters: the mean $\mu \in \mathbb{R}^p$ and concentration $\kappa \geq 0$, where $\mu$ is constrained to have unit norm.
Why not instead define the distribution for one unconstrained parameter i.e. $y := \kappa \mu \in \mathbb{R}^p$?
What is gained (conceptually or computationally or statistically) by thinking in terms of two parameters rather than one?

The von Mises-Fisher distribution is a distribution on the surface of a sphere.  To make it as easy to visualize, think of a circle.  The vM-F distribution has two parameters: the mean direction in which points are distributed on the circle, and how concentrated they are around the point on the circle in that mean direction.  You can think of this as analogous to the mean and standard deviation of a Normal distribution, or the median and scale parameter of a Cauchy distribution, or...  In higher dimensions, the parameter interpretations are the same.
In the higher dimensions, an alternative way of thinking of it is as a point on the surface of a sphere which is the center of the distribution and how "spread out" the distribution is around that point. If κ=0, the spread is such that the points are uniformly distributed on the surface of the sphere, but as κ gets large, the distribution concentrates around the center point. You can think of these two features of the dist'n independently of each other, and it is often helpful to do so.
As @Scortchi - Reinstate Monica points out in response to a comment below:
"... there are plenty of applications in which $\mu$ is the parameter of interest & $\kappa$ a nuisance parameter - testing hypotheses about $\mu$, making point & interval estimates of $\mu$, are the aims of the study. It's hard to think of an application where the components ($\kappa \mu_1,\kappa \mu_2, \dots, \kappa \mu_p$) would be of separate interest."
