I did actually do two batches with the two methods within a few weeks of asking the question.
Just cooking it all together, then removing the skins and seeds with a food mill, is definitely a lot faster. Removing the skins first then pureeing, so that you only remove the seeds later... well, though it is easy to remove the skin from each individual grape, time-consuming is an understatement unless you're making small quantities. It took me a few hours for (if I recall correctly) 10-15 pounds of grapes.
Including the skins has a moderate effect on the texture - it's slightly thicker and more opaque. You get the feeling there's just more stuff in it. As long as you're cooking it fairly long, though, there's not a very noticeable difference in flavor. On the other hand, if you're using a recipe with pectin instead of cooking it down until it sets, I definitely prefer having the skins in, for both flavor and texture.
So there's some personal preference involved; I do rather like the texture with the pureed skins. But if you're doing large quantities, and the grapes are on the small side, and you don't have help, you might want to skip it.