Dairy is a bit iffy for me. Casein, a water-soluble dairy protein, is found in butter (although it is found to a greater extent in milk). This protein is structurally similar to gluten which leads to all manner of nasty inflammatory symptoms -- such as acne.
When I consume a lot of dairy, I break out a bit.
When I consume a lot of gluten, I break out a lot.
Some people are unusually sensitive to one or the other, but most -- especially Europeans -- are more sensitive to gluten. My n=1 experience is that if I want to avoid acne altogether, I have to limit dairy AND nightshades. You may want to take those into account, too. The lectins present in them can cause inflammation, among other effects.
Conventional vs. grassfed makes no difference with respect to casein, either. Both contain casein, as it is a totally normal protein found in dairy.
Ghee gets rid of most of the nonfat components of butter, so most of the casein is gone. This is the best option for anyone who wants to cook with "butter" but finds that they're sensitive to dairy.