They are basically the same but, for some reason, many prefer to use the more exotic name ghee as a generic for plain old clarified butter, not to mention buying something that they can easily make at home fresh for less than store-bought - and probably get higher quality. There are differences.
Some traditional ghee is made from butter that was churned from milk which was first clabbered, giving the ghee a distictive sharp/sour taste. This is much like the difference between cream and creme fraiche.
Generally, recipes for making your own ghee direct that you heat the butter until the butter solids turn golden or light brown, which gives flavor to the ghee. In clarifying butter, you don't allow the solids to color so there isn't any of that slightly "burned"or caramelized flavor.
The purpose of removing the milk solids is that they burn at a much lower temperature than the butterfat. The butterfat can be heated to a much higher temp allowing you to use it as a cooking fat for sautéeing. Both have distinctive flavors which they impart to the foods cooked in them.
It's pretty much up to you which flavor you prefer. You can make your own version of either. There are many recipes on the internet and certainly, for ghee, at a substantial savings over the purchased version.
Ghee is something like clarified butter on steroids. Ghee is slightly thicker than clarified butter and has a flavor that is richer and more complex. Like clarified butter, ghee begins by melting unsalted butter and then slowly cooking it until the milk solids separate and turn brown. But ghee goes a step beyond clarified butter, continuing to be slowly simmered until all the butter’s water has evaporated and the remaining liquid becomes a beautiful golden color. (Clarifying butter takes about 10 minutes; cooking ghee takes around 15-18 minutes.) You have to cook it over very low heat and watch it carefully so that it doesn't burn. It's done when it stops making a soft crackling sound and becomes silent. Strain it carefully to separate it from all of the milk solids.
Ghee is central to the complex and sophisticated cuisines of India. But it's also great spread on toast or used to scramble eggs. When properly made, ghee needs no refrigeration and will keep for up to a year. It is considered to have powerful medicinal properties--it warms, nourishes and lubricates the body's tissues--and it will improve with age. But it is such a wonderful substance with so many uses that it tends to go quickly.
My advice: If you are doing some Indian cooking, don't settle for using clarified butter. Go for the real thing--it's worth it. And it's so easy to make that there is no reason to spend the extra money to buy ghee in a store. (Also, I have heard that sometimes commercial ghee is cut with other less expensive oils but sold as an all-butter based product.)