Does the Rinnegan have different abilities depending on the user? 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.

From the Rinnegan wiki article:


  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.


Hence Madara- 


  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.