The commonality of the etymology and initial meaning of these words is obvious. But in Christianity and Islam, they have acquired slightly different functions.
Jewish and Christian Satan
First of all, let's recall a little about who Satan is. Yes, it is already clear that this word is used as a proper name, in the singular. Satan is alone. It is difficult for us even to say how this word could be inclined in the plural: several Satans or Satan? No, Satan is a specific personal name.
In some of the ancient books of the Old Testament, the Hebrew word "Satan" did not denote a category of some creatures, but some qualities of a person himself harmful to faith. In the Church Slavonic and Russian translations of the Bible, there is no Satan as a common noun, although in a figurative (abusive) sense it has long been used in our everyday speech in relation to a person who interferes with some pious actions or the mood of other people by his behavior. And this meaning is very close to the original Hebrew.
But already in the Books of the Prophets (Zechariah, Job), Satan is a specific character, usually interpreted as a dark, fallen angel tempting the faith of the righteous. Theologians believe that Satan is also mentioned in some other books (Isaiah, Ezekiel), but is not named so or is called allegorically (for example, the Lightbringer, aka Lucifer).
In the Gospel, Satan tempts Christ, and in the epistle of the Apostle Paul to the Ephesians, Satan is called "the Prince of the Lord in the air." From here comes, apparently, the Christian idea of Satan as the leader of the dark, hellish army opposing God. It found its fullest expression in the Apocalypse of John the Theologian.
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Functions of Islamic shaitans
The Islamic concept of shaitans goes back to the common noun. From the personification of the bad qualities of a person, the word "shaitan" has passed to designate a whole category of spirits. Shaitans, according to Muslim belief, are many. Moreover, each person has his own shaitan, like his own angel. This concept, however, has its own correspondence in Christian views, only Christians call such a personal dark angel not Satan, but a demon or demon.
The work of the shaitans is simple, but troublesome: incite a person to commit impious acts, to doubt his faith, to prevent him from performing prayers and godly deeds. Here is one of the indications of the functions of the shaitans in the Qur'an. “Shaitan threatens you with poverty and orders you to do abominations,” says Surah al-Bakara (2: 268; semantic translation by Elmir Quliyev).
Witchcraft, magic, magic - all this is just the art of the shaitans, which they teach people so that they stop worshiping Allah and relying on His mercy. Belief in Allah and thorough observance of the instructions of the Koran are the main guarantee against the machinations of the shaitans. The Almighty says in Surah al-Hijr: “We have protected him [man] from every shaitan” (15:17).
The nature of the shaitans and their relationship with their head - Iblis
The essential difference between shaitans and angels is that shaitans have a material nature. They were created by Allah from a smokeless flame (15:27). However, if they do not wish, then a person can neither see, nor hear, nor touch. In this they are somewhat like angels that they can imitate. Shaitans can take on different guises, including the guises of people. The most insidious guise of a shaitan is when he takes the form of a real angel. Of course, such a shaitan's disguise cannot deceive a righteous man, but a person who is very sinful, unstable in faith, can be tempted by this.
Shaitans are one of the varieties of jinn. In the pre-Islamic era, the Arabs called jinn any spirits in general, and they could be both evil and good. After the emergence of Islam, all jinn were written into evil spirits. Not every genie is a shaitan, but every shaitan is a genie. Djinn can harm a person on their own initiative. Shaitans are the most powerful, malicious and cunning of the genies, directly serving Iblis.
Iblis corresponds to Christian Satan as a fallen angel among Muslims. He is the leader of the host of countless shaitans. In places in the Qur'an, the word "shaitan" is meant specifically Iblis. This is due both to the undesirability of mentioning the proper name of the Angel of Evil, and to the fact that initially the personality of Iblis and his servants as personifications of evil were not clearly differentiated. This has already been done by Muslim theologians in the centuries following the writing of the Qur'an.