Ghee and butter come from the same place but are two different things and the differences are much bigger than one might imagine. Ghee is butter without any water left in it and with all the milk solids removed. Simple sounding as it may be, this clarification significantly changes the properties of the product in question.
Namely, there are no milk solids left so ghee is completely lactose and casein free. This makes ghee suitable for many people who are sensitive to dairy products and would otherwise have to put up with feeling sick after having their morning toast. So in terms of tolerability ghee wins hands down.
Ghee is also superior when it comes to cooking since it has a smoke point of 250°C whereas butter's smoke point is 150°C. What this means is that you will not burn the fat when you cook with it. Not only will your dishes taste less bitter but also you won't be introducing harmful oxides into your body. So overall, for frying, roasting and other methods ghee is clearly preferable.
Ghee and butter do not really differ nutritionally except for the fact that there are more fat soluble vitamins (such as A, E and K) as well as butyrate (which is said to promote gut health and decrease inflammation) in ghee than in butter. So in terms of nutrients ghee wins a little bit.
Is ghee better for you? Overall, yes. Most of the time it is healthier than butter for most people. However, there is an approach to everyday food choice that I came across and that changed my perception of my diet completely.