Quranic Law vs Sharia Law
But if they turn away from you, your only duty is a clear delivery of the Message. (Qur'an 16:82)
Say: 'Whatever good betides you is from God and whatever evil betides you is from your own self and that We have sent you to mankind only as a messenger and all sufficing is God as witness. Whoso obeys the Messenger, he indeed obeys God. And for those who turn away, We have not sent you as a keeper.' (Qur'an 4:79-80)
And tell my servants that they should speak in a most kindly manner. Verily, Satan is always ready to stir up discord between men; for verily; Satan is mans foe.... Hence, we have not sent you with power to determine their Faith. (Qur'an 17:53-54)
Say: "Obey God, and obey the Messenger, but if ye turn away, he is only responsible for the duty placed on him and ye for that placed on you. If ye obey him, ye shall be on right guidance. The Messenger's duty is only to preach the clear (Message). (Qur'an 24:54)
And so, invite them your task is only to invite; you cannot compel them to believe. (Qur'an 88:21-22)
You are not a keeper over them. But if they turn aside from you, for We have not sent you to be a keeper over them; your task is but to preach. (Qur'an 42:48)
Obey God then and obey the Messenger, but if you turn away, for the duty of Our Messenger is just to deliver the message. (Qur'an 64:12)
And when they hear vain talk, they turn away there from and say: "To us our deeds, and to you yours; peace be to you: we seek not the ignorant," It is true thou wilt not be able to guide whom thou loves; but God guides those whom He will and He knows best those who receive guidance. (Qur'an 28:55-56)
Assuredly, We have sent down the Book to you in right form for the good of man. Whoso guided himself by it does so to his own advantage, and whoso turns away from it does so at his own loss. You certainly are not their keeper. (Qur'an 39:41)
And they ask, "When shall the promise be fulfilled if you speak the Truth?" Say, "The knowledge of it is verily with God alone, and verily I am but a plain warner..." (Qur'an 67:25-26)
As seen from the verses above; much involves obeying the Prophet but also emphasizes the Prophet's limited authority, something that most Muslims from the Sunni and Shia sects do not recognize. According to them, their rulers have the authority to punish people for what they consider sins like drinking alcohol, eating pork, not fasting during Ramadan, etc.
The Quran meanwhile focuses on crimes against one another such as stealing, killing, the slandering of women, oppression (such as usury), etc. The Quran gives the right to fight against those who fight you but not to transgress beyond that. It also gives people the right to defend themselves against evictions from their lands or property. There is no talk about punishing people for something that does not concern somebody else's right.
Adultery (also includes fornication; ie any sexual activity outside of marriage) is the only place where the Quran diverted from this due to the fact that adultery affects another party. The Quran sees adultery as affecting the other partner in a marriage. It’s a betrayal and a breaking of an oath. Strict standards are placed with regards to adultery, however, such standards become quite lenient when it comes to punishing slanderers of women. Adultery needs four witnesses, yet one who slanders can receive punishment simply from claiming adultery took place, even without the need of having four witnesses. It’s clear that the verse made it very difficult to implement punishment for adulterers, but very easy to implement on slanderers.
The Quran wanted to end the slandering of women due to the fact that honor killing traditions were very common in tribal Arabia (and still is). The Quran gave a symbolic penalty for adultery since any person would lie even under oath when facing lashes. The Quran allows people to utter disbelief when under duress (16:106), so God knows that people will lie to avoid lashing. A slanderer meanwhile needs to produce four witnesses or the penalty of flogging can fall on him. Which means he will have to lie when asked even if he saw something. The truth here is not relevant. Even when a spouse has witnessed something he has to swear but the other party can also swear. Most people as we know will simply lie to avoid the punishment since the punishment of this kind has torment in it. The person can simply repent instead. To make up for this we have the verses in chapter 4 which is the more practical way of dealing with such. The Quran here established that adultery is a major sin but made the penalty for it impossible to practice. It's four times easier for a slanderer to get himself in trouble than an adulterer.
Further reading of the verses about the Zani (male adulterer) and Zania (female adulterer) show us that the issue came up concerning slandering of one of the Prophet’s wife presumably.
Nonetheless, the Quran cannot order the Prophet to punish people for sins, that God's job. The Quran gave people the right and freedom to disbelieve, let alone sin. How the Quran understands sin is very different from how the mainstream Muslims understand sin.
But the texts of hadith claimed otherwise. The Hadith texts are claimed by Muslims to be the word and deeds of Muhammad that were compiled some two centuries after Muhammad's death. Muslims claim that Muhammad received two revelations from God, one is the Quran the other they claimed are his personal saying and actions collected by later Islamic authorities (oral traditions). Muslims believe that this secondary source has legal authority, it is through this text that almost 90% of so-called "Sharia law" is derived from. In such texts, it is claimed that the Prophet was ordered to fight the people till they acknowledge monotheism and to order the execution of those who that have been guilty of apostasy. This is why you see places where Sharia law is implemented filled with such concepts like searching cars for alcohol or flogging people for not wearing "proper attire", etc. None of this should concern anyone but it has become a punishable sin. Muslims believe that God only punishes those who did not get caught and punished in this life. Muslims claim that once punished the sin is essentially erased for good. You will not find such a concept in the Quran. God punishes in endless ways and does not need humans to punish for him. Muslims introduced this concept to coerce people into accepting their religious dogmas.
One thing is crystal clear from all this; the Quran's take on human authority and freedom is radically different than how the traditional Muslim sects understand it. Therefore the biggest difference between a state governed by the Quran and an "Islamic" state, is the state's authority over the masses.