There's a difference between the Rinnegan Madara and Hagoromo have and the Rinnegan Sasuke has.
Madara's Rinnegan and Hagoromo's Rinnegan were awakened naturally. Sasuke's Rinnegan was given to him from the Yin chakra of Hagoromo. Therefore, his Rinnegan works like a Mangekyo Sharingan. When he uses it too much, he has to close it and let it recharge. Madara and Nagato do not have to recharge their eyes.
If I recall, Hagoromo's Rinnegan was considered a Kekkei Mora whereas Madara's is considered a KKG. This would make sense considering Hagoromo's feats of the Rinnegan far surpasses Madara's as Madara cannot exhibit Yin-Yang Release even close to the level that Rikudou Sennin can.
Sasuke's Rinnegan was naturally awakened as well, the only difference is that Hagoromo gave him his own chakra which was processed and formed a change in his eyes as a result. It's still considered his Rinnegan in the Databook, so it was naturally awakened like Madara's was.
Difference is, is that because he has Rikudou's Chakra and Indra's Chakra, he uses them both simultaneously which allows him to use the Rinnegan and EMS both and forms his Tomoe Rinnegan. With this, he can actually use his Rinnegan to further amplify his Sharingan techniques on a much larger scale and actually use his Rinnegan to complement his Sharingan's abilities such as when he used Outer Path and formed a Gedo Statue using his PS.
Sasuke only has to recharge his eye due to the fact that he was inexperienced with the Rinnegan and the fact that his techniques drains a lot of his chakra, mostly Ameno. When his chakra is drained, he's restricted to a normal Rinnegan which still enables him to use techniques such as Ameno as shown in Gaiden, only problem being that he cannot use his Sharingan's abilities through that Rinnegan until after he recovers enough chakra.
Sasuke has an incomplete Rinnegan, that's why it has 6 tomoe. The tomoe are from the two Sharingans of him and Itachi combined.
More accurately, he has an incomplete RinneSharingan, the RinneSharingan having the Rinnegan and 9 Tomoes in 1 eye whereas Sasuke has the Rinnegan and 6 Tomoes in 1 eye. The Juubi is also shown to have the same eye as Sasuke does, so it's certainly a superior eye compared to the Rinnegan.
Madara's Rinnegan is true, not a half Sharingan, half Rinnegan incomplete eye.
Not really. Sasuke's eye possesses both the Rinnegan and the EMS and with that, he can use the Rinnegan to amplify his Sharingan and actually use the Sharingan's abilities through that eye which makes it superior to Madara's.