If the cost is about the same, I would do the addition to the house -- it adds to the value of the property, while a shed is more likely to be a liability down the road. You also have the issue of going outdoors in bad weather and heating/cooling a shed, things that are more easily controlled with a house addition. If you like the sense of separation, then put in a good door and some sound proofing. I always think that I should put a screen door on my sewing room -- so I can hear what's going on with the kids but keep the pets off my fabric, but I can definitely see the advantage of tuning everything out.
That said, I think that in our area, a home addition would cost more than putting up a shed-- by quite a bit.