I haven't noticed a difference between white sugar, brown sugar or honey. However, this doesn't mean there isn't one. Sometimes I'm not very observant. The one I used brown sugar in got hot but I used spicy EOs in it which I think is what contributed to the heat. A couple of people have mentioned confectioner's sugar makes their soap feel silkier because of the cornstarch.
I like to use simple syrup. Just bring equal parts of water & sugar to a boil, stirring. Let it simmer for a minute or so until the sugar is dissolved. I keep a jar of it in the fridge, and then add a TB or two to my lye/water mixture. That way, I don't have to worry about whether the sugar will dissolve or caramelize in the lye.
There's a soaping 101 video where she used I think maple, brown and white(?) all together in a soap, it affected the colour. Nice video, I don't think she noted any difference in the attributes between them. I switched to confectioners which I add to my oils before adding the lye solution. It's the easiest way I remember to put sugar in
This is just my opinion but I base it on the recipe. If I had a bubbly value of 23, I may not add sugar if I had castor oil in the recipe (and I normally do include castor). Castor helps to boost and stabilize lather. It does add a little creaminess, too. I prefer to see a range of 22-26 for my recipes but I don't always achieve this depending on the recipe. I think the lowest bubbly value has been 17 in one of my recipes.
It might be helpful to make a batch with sugar and one without to see if there is a difference.