I think this is an instructive question. Here are some partial answers.
A category fibered in groupoids whose fibers are sets (e.g. no automorphisms) is a presheaf. Strictly speaking, I mean equivalent to the fibered category associated to a presheaf. This truly follows from the definitions, and is a good exercise to do when one is learning the basic machinery behind stacks. Similarly, a stack which is equivalent to a presheaf is a sheaf (i.e. the descent condition collapses to the sheaf condition; this is a little harder but still tautological). And a pre-stack with no automorphisms is a separated pre-sheaf.
The other claim is more interesting and not tautological, and reflects the fact that the diagonal of a morphism of stacks is way more interesting than in the case of schemes. For instance, an algebraic stack is DM iff its diagonal is unramified. (I believe this is in Champs Algebriques, but there's a nice discussion in Anton's notes from Martin Olsson's stacks course.
The point is that the diagonal of a stack carries information about automorphisms of the objects it parameterizes. So for instance the condition (in the definition of a stack) that the diagonal is representable is equivalent to the statement that Isom(X,Y) (and thus Aut(X)) is representable by an algebraic space. Also tautological is the statement that the diagonal being unramified is equivalent to the statement that there are no infintesmal automorphisms (e.g. non-trivial automorphisms of an object over $k[\epsilon]/\epsilon^2$ which reduce to the identity map over $k$). So here it is now clear where one uses finiteness: the $k[\epsilon]/\epsilon^2$ points of a finite groups scheme are the same as the $k$ points; on the other hand this fails if the automorphism scheme is, say, $\mathbf{G}_m$.
Finally, while the tautological answer above answers your question, it is instructive to see how automorphisms cause $M_g$ (moduli of genus g curves) to not be representable. Let H be a hyperelliptic curve given by $y^2 = f(x)$ defined over a field $k$. Then the curve $H_d$ given by $dy^2 = f(x)$ is not isomorphic to $H$ over $k$ if d is not a square in k. Call this a `twist' of H; in general twists of a variety X are given by the Galois cohomology group $H^1(G_k,Aut X)$ which is non-zero in non-trivial situations (alternatively you can use torsors and etale cohomlogy), and a generic hyperelliptic curve has automorphism group $\{\pm 1\}$. Thus H and $H_d$ give two different $k$ points of $M_g$ which become the same point over a finite extension; thus $M_g$ fails the sheaf condition in the etale topology, (and so in general existence of automorphisms causes, for cohomological reasons, failure of your moduli problem to even be a sheaf).
Last comment (to clarify others' comments): fine moduli space should certainly allow algebraic spaces for a correct answer to your question.
An example is Deligne's theorem on the existence of good notion of quotient $X/G$ of a separated algebraic space $X$ under the action of a finite group $G$, or relativizations or generalizations (with non-constant $G$) due to D. Rydh. See Theorem 3.1.13 of my paper with Lieblich and Olsson on Nagata compactification for algebraic spaces for the statement and proof of Deligne's result in a relative situation, and Theorem 5.4 of Rydh's paper "Existence of quotients..." on arxiv or his webpage for his generalization.
Note that in the above, there is no mention of DM stacks, but they come up in the proof! The mechanism to construct $X/G$ (say in the Deligne situation or its relative form) is to prove existence of a coarse space for the DM stack $[X/G]$ via Keel-Mori and show it has many good properties to make it a reasonable notion of quotient. Such quotients $X/G$ are very useful when $X$ is a scheme (but $X/G$ is "only" an algebraic space), such as for reducing some problems for normal noetherian algebraic spaces to the scheme case; cf. section 2.3 of the C-L-O paper. I'm sure there are numerous places where coarse spaces are convenient to do some other kinds of reduction steps in proofs of general theorems, such as reducing a problem for certain DM stacks to the case of algebraic spaces.
Also, Mazur used a deep study of the coarse moduli scheme associated to the DM stack $X_0(p)$ in his pioneering study of torsion in and rational isogenies between elliptic curves over $\mathbf{Q}$ (and these modular curves show up in numerous other places). But those specific coarse spaces are schemes and can be constructed and studied in more concrete terms without needing the fact that they are coarse spaces in the strong sense of the Keel-Mori theorem, so I think the example of Deligne's theorem above is a "better" example.
Best Answer
An elementary example everyone should know is
BG=[*/G].
Here * is a point and G is an algebraic group. Its sections on a scheme X form the groupoid of principal G-bundles on X. Because principal G bundles are locally trivial, the coarse moduli space is a point.
If M is a stack, it determines a presheaf \pi_0(M), whose sections are the isomorphisms classes of objects of M. One way to think about the coarse moduli space is as a space representing the sheafification of this presheaf in whatever topology you are working with.
Starting from this example, you should also be able to work out the issues surrounding [X/G]. In fact, it's good to revisit BG after working out the general case. But I won't do it here since you asked for one example only. In fact, maybe I should have said
*= Spec(Q)
and
G=GL_2/Q.
I learned about stacks in the days before all these nice books, so I'm not sure about a reference for anything I write about them. But the statements here should all be pretty clear from the definitions.