Every element $\alpha$ in $PGL(2, \mathbb R)$ is represented by a matrix $A \in GL(3, \mathbb R)$; two such matrices $A_1, A_2 \in GL(3, \mathbb R)$ represent the same element $\alpha$ if and only if $A_1 = kA_2$ for some nonzero $k \in \mathbb R$.
It seems to me that for each $A \in GL(3, \mathbb R)$, we can choose $k = \frac{1}{(\det A)^{1/3}}$, so that $\det (kA) = 1$, and therefore $kA \in SL(3, \mathbb R)$. Therefore there is a "natural" isomorphism between $PGL(2, \mathbb R)$ and $SL(3, \mathbb R)$.
Is there any reason not to identify $PGL(2, \mathbb R)$ with $SL(3, \mathbb R)$? Is there a sense in which this identification obscures some important distinctions we should care about?