One of their primary abilities is the ability to know a Servant's True Name and all their skills. in Holy Grail Wars like the Fuyuki, Trifas and Moon Cell ones were knowing a Servants identity allows a Master a greater advantage in knowing their weaknesses, there are limits to this skill which are shown with both Jeanne and Amakusa
True Name Discernment (B Rank): When summoned as a Ruler, status information such as the true identity, Skill and Parameters is automatically revealed when she directly encounters a Servant. However Servants with hiding capacity depend on her luck level to bypass this.
Source: Jeanne d'Arc (Ruler) > Abilities > Class Skills
True Name Discernment (B Rank): As a Ruler class, he gains this skill. Status information such as the true identity, Skill and Parameters is automatically revealed when he directly encounters Servant. There are certain limits, Servants with hiding capacity such as Saber of Red's Noble Phantasm Secret of Pedigree can mask out her identity, however it depends on his luck level to bypass this.
Source: Amakusa Shirou Tokisada (Ruler) > Abilities > Class Skills
as mentioned with both Mordred's Helmet, "Secret of Pedigree" hides her identity from them and requires a Luck check which Jeanne's being Rank C and Amakusa's being Rank B. it's possible that B+ to EX might be able to negate Mordred's Nobel Phantasm however quick skimming the Grand Order Rulers none of them have True Name Discernment.
Rulers also have the God's Resolution Skill which allows them to have Command Spells. Jeanne's being Rank A allows her 2 Command Spells per Servant and can not repurpose used ones. also she can not affect Heroic Spirits if they have given up their existence as a Servant but remained in the world
When the nameless vampire previously known as Lancer of Black and Darnic Prestone Yggdmillennia abandoned his existence as a Servant, this skill was no longer able to restrain him.
Amakusa did have this skill in the past but it is unranked because he has lost it with his Command Spells being his own as being a Master and the ones that were given to him by the other Masters of Red, very likely acting in a similar manner to a supervisor from the church like Risei Kotomine
Finally Rulers also have a high Magic Resistance which is always a very desired skill in the Holy Grail Wars with normally Sabers having the highest rank. Amakusa's is at the equivalent to a string Saber's while Jeanne's is the highest rank, yet Sacraments like from the church are not affected. we can probably take Amakusa's Rank A as being the base as generally Saints are what qualify to be Rulers but Amakusa was an aberration due to Einzbern Cheating while with Jeanne it's probably boosted to EX because of who she is
In addition to the Magic Resistance of the Saber Class, she demonstrates a high anti-magic from her unwavering piety.
In addition to their class Skills, Rulers are supposed to be neutral during a conflict with Amakusa probably not being neutral due to how he was summoned as the Grail generally summons a Ruler, not a Master. they are also supposed to make sure Heroic Spirits follow the laws placed upon them. ie "the dead not leading the living"
A Ruler is also summoned to preserve the laws of Heroic Spirits to ensure they won't be violated, such as the law of "the dead not leading the living", touted as the greatest law of Heroic Spirits, and to maintain human history by preserving the illusion behind the Age of Gods so that people do not know about the truth behind it.
Source: Ruler (Second Paragraph)
while it was release much earlier, a probable reason why no Ruler is summoned to stop Gilgamesh or Angra Mainyu in Fate/Zero and Fate/Stay Night is probably due to the damage caused to the Greater Grail and the system thanks to the Einzbern summoning Angra Mainyu who contaminated the Greater Grail since the 3rd war.
i will add that the wiki lists Karl der Große as having God's Resolution and True Name Discernment however his wiki page only lists their ranks and not what they effects are and not having played Fate/Extella yet i don't have first hand knowledge what their effects are, however they are at a lower level so we can assume they are not as powerful
Also as stated by Swimsuit Martha who is a Grand Order Ruler, the Grand Order has altered the Ruler Class
“It is expected that I have it, but… seeing that it is not a Holy Grail War, what I know is that even the role of a Ruler has come to be altered in due course, do you not think so?”
this would explain why when you look at the Rulers listed as being in Fate/Grand Order none of them seem to have God's Resolution and True Name Discernment