If one looks up the definition of a Deligne--Mumford stack or an Artin stack, one usually finds something like:

A DM (resp. Artin) stack is a stack $X$ satisfying [insert condition in the diagonal here] and for which there exists a representable etale (resp. smooth) surjection $U\to X$ from a scheme $U$.

On the other hand, we are also told that we should think of a DM stack as something which is locally isomorphic to $Y/G$ where $G$ is a finite group acting on a scheme $Y$, and that we should think of an Artin stack as something which is locally isomorphic to $Y/G$ where $G$ is an algebraic group acting on a scheme $Y$. This discussion is not limited to algebraic stacks: one can have a similar discussion in the context of differentiable or topological stacks.

I have found relatively little material, however, on comparing and contrasting these two styles of definitions. The definition in terms of atlases seems by far the most common in the literature. I am thus lead to ask:

Is there a particular reason to prefer the definition in terms of atlases over the definition in terms of local quotients? At what point in the theory and/or applications of DM/Artin stacks is the distinction between the two styles of definitions relevant?

(Possibly relevant subquestion, but not the main question: what is involved in passing between the two styles of definitions?)

I am actually most interested in the answer to this question in the context of topological stacks, but I would also be happy to have an answer in the algebraic context.