Somehow this question came back up after many years. I didn't think the answers really got to the essence of the situation.
Asking for a fine moduli space is asking for a several different things at once that are related but not quite the same. So not only are the nontrivial automorphisms a problem but also that isomorphisms between bundles can degenerate. I'll leave it to a licensed Algebraic Geometer to fill out the requirements for being a fine moduli space. I think it is instructive to explain that even in the simplest nontrivial example, rank 2 bundles on $\mathbb{P}^1$ the situation is already subtle. A key phenomenon is that a naive moduli space (isomorphism classes of bundles with a topology that forces holomolorphic (or algebraic) families of bundles is to give rise to a continuous maps to iso classes) typically is not Hausdorff. To see this consider rank two bundles on $\mathbb{P}^1$ (working over $\mathbb{C}$). Suppose further that the degree of the bundle is zero. By the Dedekind-Weber, Bertini, Birkhoff, Grothendieck theorem such bundle is isomorphic to a sum $\cal{O}(d)\oplus \cal{O}(-d)$ (say $d\ge 0$ to rid ourselves of some ambiguity). So you great the set of isomorphism classes is parametrized by $\mathbb{Z}_{\ge 0}$. Viewing $\mathbb{P}^1$ as $\mathbb{C}\cup \infty$ with coordinate $z$ around $0$ and $1/z$ around infinity you can write the transition function for such bundle as
\begin{pmatrix} z^{-d} & 0 \\ 0 & z^{d} \end{pmatrix}
which admits the deformation \begin{equation}g_{d,t}=\begin{pmatrix} z^{-d} & 0 \\ t & z^{d} \end{pmatrix}.\end{equation}
Define a new trivialization (when $t\ne 0$) away from $\infty$ by \begin{equation} \{\begin{pmatrix} z^d\\-t \end{pmatrix}, \begin{pmatrix} 1 \\0\end{pmatrix}\}. \end{equation} This gets mapped under $g_{d,t}$ to \begin{equation} \{\begin{pmatrix}1 \\ 0 \end{pmatrix},\begin{pmatrix} z^{-d} \\ t \end{pmatrix}\}. \end{equation} This is trivialization that extends over $\infty$ (when $t\ne 0$). Thus the family of bundles with transition function $g_{d,t}$ gives the trivial bundle for $t\ne 0$ and $\cal{O}(d)\oplus \cal{O}(-d)$ for $t=0$. This shows that the closure of the trivial bundle contains every bundle. More generally the closure of $\cal{O}(d)\oplus \cal{O}(-d)$ contains $\cal{O}(e)\oplus \cal{O}(-e)$ for all $e \ge d$.