How an author approaches the short story has quite an effect on the end result. In my mind, you can always tell when a short story was, at one point, longer and then cut down here and there vs. a story that was concise to begin with. Is one way better than the other? Not really. But I do think it's important to be aware of the fact depending on how you want your story to read.
I think the biggest difference between these two types of short stories is the feel. Do you want it to feel like a mini novel or a flash in a pan? A concise story begins with a concise concept. As in, if you're already thinking about fleshing character out or multiple plot lines, it's not a concise concept.
Changing your approach to a story is an interesting thing to experiment with. Try writing one story with a definite beginning, middle and end. Then, if need be, cut it down a little so it flows. Then try just describing one single concept. Like eating an ice cream cone or something. I've tried both styles and it really does change things around. It forces you to explore the natural length of a story rather than adhering to a specific word count.