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Well there are a few differences, seeing as the Mangekyou sharingan's powers differ from each different sharingan user. Such as Kakashi's, his Mangekyous power is to manipulate space, while Sasuke's was Susuanoo. While the Rinnegan's power is the same in each user, but, in terms of power and activation, the Sharingan has four basic forms (three for most), one pronged start off, then to two prongs, then three prongs, then for the ones who can Mangekyou. While the Rinnegan's appearance begins as many different swirls around the pupil and stays that way, and while the Sharingan can be deactivated, the Rinnegan can't.