Here is a short summery of the answer:
Islam is not significantly different from the previous Abrahamic religions in their original forms. However there are important differences between Islam and the current practices of these religions (Judaism/Christianity). The objections made against the followers of Judaism/Christianity are mainly because of them not following their original convent with God. It confirms them, states the main divergences, and updates the sacred laws. Some differences in perspective between them are:
Islam emphasizes a very strong form of monotheism where God is the center of everything and there is no other (independent) force. This in addition to the fact that humans should submit to his will is the core of Islam (and where the name "Islam" comes from).
Spirituality and sacred laws are both important for Islam and in a sense Islam falls in the middle of the range between Judaism (focuses on following complicated external religious laws and has less focus on soul/spirituality) and Christianity (focuses on soul/spirituality and has less focus on following the sacred rules). The goal is to obtain growth and both "iman" and "good deeds" are required for this, neither is sufficient alone. Also both personal and social aspects are important in Islam.
There the story of creation is significantly different from the one in the Old Testament/Tanakh. This gives a different perspective regarding why God created humans and what they should do in this world.
Islam encourages thinking and understanding. "Iman" (which is often translated to "belief" in English) comes through knowledge. The opposite of "Iman" is "Kufr" (covering/hiding truth). Covering/hiding/rejecting truth (when one is aware of it) is the biggest "zulm" (i.e. injustice, cruelty). "Zulm" and "Kufr" are the central negative attributes in Quran (similar to "evil" in Judeo-Christian literature).
Regarding non-Abrahamic religions, they are typically not monotheist, and any religion which is not monotheist is strongly inconsistent with Islam's core beliefs. According to Quran, anyone who does not submit to the will of God will follow other false patrons, consciously or unconsciously, and following false patrons will lead to astray from the right path towards the "light" and the "growth".
From Quran's view, the core beliefs (there is one and only one God, there will be a judgment day, there are prophets sent by God, ...) and core required actions (Salah, Zakat, Sawm, Hajj, ...) in Islam are similar to those of previous Abrahamic religions. Quran invites them to accept Islam which is better.
The objection made regarding the followers of the previous Abrahamic religions like Jews and Christians are mainly about them not following God's orders and breaking their convent with him, e.g. Christians considering Jesus (PBUH) as God and not following the scared laws (following Paul's teachings), Jews rejecting God's prophets like Jesus and killing them, etc.
The second suras of Quran talks with them and asks them to return, e.g. in verse 2:40 and several other places Quran tells the Israelities (i.e. the children of Israel):
يَا بَنِي إِسْرَائِيلَ اذْكُرُوا نِعْمَتِيَ الَّتِي أَنْعَمْتُ عَلَيْكُمْ وَأَوْفُوا بِعَهْدِي أُوفِ بِعَهْدِكُمْ وَإِيَّايَ فَارْهَبُونِ
O children of Israel, remember the favor which I bestowed upon you, and fulfill your covenant with me and I will fulfill my covenant with you, and [only] me you [must] fear.
(This covenant, according to Quran, includes believing and accepting the prophets that God sends including Jesus and Mohammad (PBUT).)
Islam is considered as the continuation of these religions (and the final one), it is not expected to have significant difference in its teachings from the original version of previous Abrahamic religions. The intention is to confirm them, correct the mistakes that have become part of them, and update the sacred laws.
The concept of "tawhid"(i.e. a strong form of "monotheism") is very central to Islam and it's the concept in Quran that everything follows around. Everything is viewed from this perspective, there is no (independent) power in the world besides God. Clouds move by his will, rain drops by his will, trees grow by his will, birth, death, ... there is no concept of natural forces causing things to happen, everything that many people today attribute to nature/laws of physics are attributed to God. Some may say that one cannot understand Islam unless one understands this concept of the centrality of God for everything and when I am saying everything I really mean everything, and there is no other (independent) power. God is the creator of everything.
God in Quran is completely abstract and beyond human knowledge (verse 5:116), he is not like anything (verse 42:11), it is forbidden to liken God to anything, it often looks more similar to an intelligent abstract force governing everything than the person-like figure in Judaism/Christianity. At the same time, God has objectives and actively effects all events, he is not a non-participating passive force. A famous verse about God is verse 2:255 which might shed some light on Quran's perspective:
اللَّهُ لَا إِلَٰهَ إِلَّا هُوَ الْحَيُّ الْقَيُّومُ ۚ لَا تَأْخُذُهُ سِنَةٌ وَلَا نَوْمٌ ۚ لَّهُ مَا فِي السَّمَاوَاتِ وَمَا فِي الْأَرْضِ ۗ مَن ذَا الَّذِي يَشْفَعُ عِندَهُ إِلَّا بِإِذْنِهِ ۚ يَعْلَمُ مَا بَيْنَ أَيْدِيهِمْ وَمَا خَلْفَهُمْ ۖ وَلَا يُحِيطُونَ بِشَيْءٍ مِّنْ عِلْمِهِ إِلَّا بِمَا شَاءَ ۚ وَسِعَ كُرْسِيُّهُ السَّمَاوَاتِ وَالْأَرْضَ ۖ وَلَا يَئُودُهُ حِفْظُهُمَا ۚ وَهُوَ الْعَلِيُّ الْعَظِيمُ
God, whom there is no god but him, the living, the self-subsisting [eternal]. The slumber does not seize him nor sleep. His are everything in the skies and on the earth. Who is there who can intercede in his presence except as he permitted? He knows what is in front of them and what is behind them, and they don't compass anything from his knowledge except as he wills. His throne extends the skies and the earth, and he felt no fatigue in guarding [preserving] them, and he is the most high, the supreme.
لَا إِكْرَاهَ فِي الدِّينِ ۖ قَد تَّبَيَّنَ الرُّشْدُ مِنَ الْغَيِّ ۚ فَمَن يَكْفُرْ بِالطَّاغُوتِ وَيُؤْمِن بِاللَّهِ فَقَدِ اسْتَمْسَكَ بِالْعُرْوَةِ الْوُثْقَىٰ لَا انفِصَامَ لَهَا ۗ وَاللَّهُ سَمِيعٌ عَلِيمٌ
There is no compulsion in [accepting] the religion, the growth has been clarified from the obliquity, whoever rejects the despot[s] and believes in God has grasped the trustworthy hand-hold, there will be no break for it. And God all-hearing and all-knowing.
اللَّهُ وَلِيُّ الَّذِينَ آمَنُوا يُخْرِجُهُم مِّنَ الظُّلُمَاتِ إِلَى النُّورِ ۖ وَالَّذِينَ كَفَرُوا أَوْلِيَاؤُهُمُ الطَّاغُوتُ يُخْرِجُونَهُم مِّنَ النُّورِ إِلَى الظُّلُمَاتِ ۗ أُولَٰئِكَ أَصْحَابُ النَّارِ ۖ هُمْ فِيهَا خَالِدُونَ
God is the guardian [patron] of those who believed, he brings them out from the darknesses towards the light; and those who disbelieve, their guardians [patrons] are the despot[s], bring them out from the light towards the darknesses, they will be the companions of the fire, they will dwell therein [forever].
God has many names, and understanding these names is the way to know about God. This also brings us to another major difference with Judaism/Christianity. From Quran's perspective, everything in the universe worships God, sun, moon, birds, etc. The reason that humans were created was because humans are capable of knowing the names, all of the names, and this gives them the ability to worship God in a way that even Angels were incapable of. The story of creation of Adam (like verses 2:30-39, 7:11-25, 17:61-65, and 20:115-124) is significantly different from the one in Old Testament/Torah.
According to Quran, God intended form the start for Adam be sent to earth, even before his creation. The story starts by God telling the angles that he is going to put a "khalifa" (i.e. "vicegerent") on earth. They object to this by stating the shortcoming of such a vicegerent and pointing out the fact that they do worship God already. God replies that he knows what they don't know and the story follows by God creating Adam and demonstrating to the Angles that Adam is capable of knowing all of the names while they are not. God orders all angles to "Sajdah" to Adam all do so but Satan who refuses to do so, claiming to be better than Adam. For this reason God expels Satan and the animosity of Satan towards Adam and his children starts. Then Satan causes Adam and Eve to eat from the tree and they are expelled also, but unlike Satan, Adam asks for forgiveness from God and God forgives him, so the original sin of Adam is already forgiven by God (unlike the Christian version of described by Paul where the original sin is significant to their narrative about Jesus).
The story of creation is important for the reason that it gives a perspective about why humans were created and what was the goal. Also the role of Satan and what he can do (he doesn't have any control on humans, Satan's major ability is tempting humans to do wrong and he is skillful at it), unlike Christianity where Satan is considered a very powerful figure. The Christian narrative puts much more emphasis on Jesus (PBUH) and Satan and God is a less important actor in the events.
Another central concept in Quran is "Iman" which is often translated to as "belief" but has a more delicate meaning. Its converse is called "Kufr" which literally means "covering [truth]". Acknowledging truth when it is presented to someone and not covering it is very important from Quran's perspective. Therefore, unlike Christianity (which following Paul's teachings considers faith and belief alone as sufficient and the main force), in Islam "iman" which comes through knowledge (in Quran's sense) takes the center role, thinking about the universe and trying to understand and see the signs of God in everything is encouraged ("iman" comes from understanding and is not in conflict with it). But this is not considered sufficient, one has to walk towards God (by acting), to become "enlightened" and able to "see". If a person knowingly covers or rejects a truth, it will cause problems for their "heart". A person who doesn't have knowledge about something should neither accept (verse 17:36) nor reject it. Saying things that one does not know is criticized.
In a sense, one can say that Islam falls in the middle ground between Judaism and Christianity (as verse 2:143 seems to suggest). The spirituality is important but actions and following sacred religious rules are also important. In Quran, when describing those who will eventually "prosper", the word "iman" always comes with "good deed".
The oneness of God and the submission of humans to his will are the major themes of Islam and the reason it is called so, i.e. "[complete] submission [to the will of God]".
If you want to have a deeper understanding of the central concepts of Quran, I think having a look at Toshihiko Izutsu's books, particularly "Ethico-religious concepts in the Quran" and "God and man in the Quran" can be helpful.
Regarding other (non-Abrahamic) religions, obviously there are major differences. Most don't even believe in monotheism and believe in independent forces in the universe besides God, and that would already make them strongly inconsistent with Islam. From Quran's perspective, this is the most important truth a human should know, and the "heart" of anyone who doesn't understand and "see" this truth has severe problems.
A few unique facts about Islam:
The main miracle of the prophet was Quran, a book. Therefore it is timeless and accessible to people centuries after the time of the prophet (unlike other miracles where one needed to be present at the time/place to see them),
Muslims consider Quran to be the word of God, each word and sentence is chosen by God himself, so in a sense it is God speaking with people directly.