The pattern of the Rinnegan on Madara and Sasuke are different. Looks like Sasuke's Rinnegan have different abilities compared to Madara's. Likewise, Mangekyou wielders have their own special abilities.
Madara's eyes were the first to awaken the Rinnegan. He then transplanted them into Nagato. Obito took one of the Rinnegan eyes and transplanted into himself.
The Rinnegan offers a vast array of techniques:
Absorption Soul Technique Amenotejikara Animal Path Asura Attack Asura Path Banshō Ten'in Blocking Technique Absorption Seal Chakra Receiver Manifestation Chibaku Tensei Demonic Statue Chains Deva Path Flaming Arrow of Amazing Ability Genjutsu: Rinnegan Human Path Indra's Arrow Limbo: Border Jail Naraka Path Outer Path Outer Path — Samsara of Heavenly Life Technique Preta Path Shinra Tensei Six Paths Technique Six Paths Ten-Tails Coffin Seal Six Paths of Pain Six Paths — Chibaku Tensei Six Red Yang Formation Summoning Rinnegan Summoning: Demonic Statue of the Outer Path Tengai Shinsei
First we look at Nagato's case.
Nagato inherited both the eyes through transplantation. Meaning: those weren't his original eyes. He was capable of using several of the above mentioned techniques.But not all. From Nagato's wiki article (emphasis by me):
Though he was not the eyes' original owner, being an Uzumaki allowed him to use the Rinnegan to their full potential
Obito took one of Nagato's eyes as a replacement. He was seen using several techniques too. From Obito Uchiha's wiki article (emphasis by me):
Although he used only one eye and was not its original owner, Obito nevertheless gained a great deal of power from it. He is capable performing all of the Six Paths Techniques with the Rinnegan, but he is only ever seen using abilities of the Outer Path: he can summon the Demonic Statue of the Outer Path, create chakra chains to restrain targets, and use Outer Path — Samsara of Heavenly Life Technique to revive the dead in exchange for his life.
Obito and Nagato were seen to have almost the same set of Rinnegan skills (Rinne-Rebirth, Six Paths of Pain Techniques, etc.), although Nagato was better experienced with their use. From the above, it is safe to presume that the Rinnegan's abilities don't have different set of Rinnegan-based skills. Since Nagato and Obito aren't the original owner(s) of the eyes, there is a certain degree of extent to which they can use the Rinnegan eyes.
The various abilities of the Rinnegan remain intact upon being transplanted into another individual, but only when both eyes are wielded by the original owner can their full power be accessed.
Because Madara was the eyes' original owner, only he could use them to their fullest power.
This also explains how he was able to use Limbo, as he was the real owner and knows techniques of the Rinnegan better.
Coming to Sasuke's Rinnegan, it is practically the same as the standard Rinnegan .
Conclusion: The Rinnegan abilities don't differ from shinobi to shinobi. It is kinda like, the original owner has the complete mastery over it whereas the ones who borrow it have a piece of the action.
Note: Rinnegan and Rinne-Sharingan are not the same.
Sume-mase's answer is wrong because :
- Kaguya and Sasuke's dojutsu aren't the same. Kaguya has the Rinne-Sharingan while Sasuke has a Rinnegan (evolved from a Sharingan).
- Sasuke having the tomoe pattern on his Rinnegan has nothing to do with Kaguya. If it had been like that , Madara's Rinnegan would also have the tomoe pattern.
Sasuke has a unique Rinnegan from my own point of view. I think Sasuke's Rinnegan is somehow between the usual Rinnegan and the Rinne-Sharigan. Here are my reasons:
First the person that ever awakened the Rinnegan was Hagoromo Otsustuki which is basically a mutation of Rinne-Sharigan and Byakugan which are his mothers original Doujutsu and Madara obtaining power from the combination of Indra and Asura's Chakra created Hagoromo's Chakra thus giving him the Rinnegan. The Rinnegan when awakened manifests in both eyes, if you recall after the battle between Madara and Hashirama, Madara died but he already foresaw his losing to Hashirama and made a time activating Izanagi at the cost of losing the sight in one eye but when he manifested the Rinnegan he regained sight in both eyes, therefore the eyes Sasuke has is not a Rinnegan and also not a Rinne-Sharigan but a mutation because he activates it in one eye and it looks like Rinne-Sharigan which has just one eye and he has six Tomoe in his Rinnegan while the Rinne-Sharingan has 9 tomoe i think the reason why sasuke does not have the full Rinnegan is because Hagoromo shared his power between him and Naruto and since he did not get Naruto's DNA or cells into him but a much tinkered cell of Hashirama perhaps that is why his Rinnegan mutated.