This is one of those things that you might not think makes a huge difference, but it totally does.
Room temperature butter still has structure. And with most cookies, you use the creaming method, where you beat the butter and sugar, right? Well, what you're doing here is, those sugar crystals actually cut up the butter, making tiny holes in there. These holes, although you might not realize it, serve 2 purposes. First, they help the other ingredients incorporate faster. And second, some of them will actually remain throughout the mixing process. And these holes give good "starters" for the baking powder to fill with gas, making a better texture for your cookies.
Melted butter will behave completely different. The water in butter will dissolve the sugar, which makes it harder to incorporate. And, the melted fat will coat the flour granules, which will make an extremely tender cookie, because it prevents the flour from making enough gluten. So your cookies at best will be a bit flat, and at worse will fall apart with no structure at all.
The room temperature butter makes all the difference, texture-wise. But if you just want "okay" cookies quickly, then melted butter will do in a pinch, since the flavor isn't really affected. When you're wanting those superior cookies, though, definately take the time and soften the butter at room temp until its soft, but not melted.