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The main advantage of real hair is that it is far easier to look after. There is no need to treat it any differently to your own hair and it looks and feels so natural that even a trained eye cannot tell you are wearing them (when they are expertly applied). When left to dry naturally they dry with the same natural kink/wave that most people's hair dries with, meaning it blends easily without even the need for styling.
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Another major advantage is that there is no restriction with styling. You can use straightening irons and curling tongs freely, which is not the case with synthetic hair.
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You also have the option of dying the real hair, which is not possible with fake hair. For these reason the celebrities usually opt for the real hair.