The Difference between Islam and Iman
Question

Allah, may He be Exalted, says (interpretation of the meaning):
“So We brought out whoever was in the cities of the believers. And We found not within them other than a [single] house of Muslims.” [Adh-Dhariyat 51:35-36]
What is the difference between the believers and the Muslims, and who are higher in status?
Praise be to Allah.The difference between Islam and Iman ( faith) is one of the issues that the scholars have discussed at length in the books on `Aqidah (belief) . To sum up what they stated concerning that: if one of these two terms is mentioned on its own, then what is meant is the entire religion of Islam, in which case there is no difference between Islam and Iman.
But if these two terms are mentioned together in a single context, then what is meant by Iman is inward deeds, which are actions of the heart, such as belief in Allah , may He be Exalted, loving Him, fearing Him, putting one’s hope in Him, may He be Glorified and Exalted, and being sincere to Him alone.
As for Islam, what is meant thereby is outward actions that may be accompanied by faith in the heart, or they may not, in which case the doer is a hypocrite or a Muslim whose faith is weak.
Shaykh Al-Islam Ibn Taymiyah (may Allah have mercy on him) said:
“The word Iman is sometimes mentioned on its own, not in conjunction with the words Islam or righteous deeds, or any other words. And sometimes it is mentioned in conjunction with the word Islam, as in the hadith of Jibreel, in which [Jibreel asked the Prophet (blessings and peace of Allah be upon him)]: “What is Islam ?… What is Iman?”; and the verses in which Allah, may He be Exalted, says (interpretation of the meaning):
{Indeed, the Muslim men and Muslim women, the believing men and believing women.} [Al-Ahzab 33:35]
{The bedouins say, `We have believed.’ Say, `You have not [yet] believed; but say [instead], “We have submitted,” for faith has not yet entered your hearts.’} [Al-Hujurat 49:14]
{So We brought out whoever was in the cities of the believers. And We found not within them other than a [single] house of Muslims.} [Adh-Dhariyat 51:35-36]
When Iman is mentioned in conjunction with Islam,
The word Islam refers to outward deeds, such as uttering the twin testimony of faith, prayer, giving Zakah, fasting and Hajj.
And the word Iman refers to what is in the heart of belief in Allah, His angels, His Books, His Messengers and the Last Day.
When the word faith is mentioned on its own, it includes Islam and righteous deeds, as in the hadith about the branches of faith : “Faith has seventy-odd branches, the highest of which is saying La ilaha illallah (there is none worthy of worship except Allah) and the least of which is removing a harmful thing from the road.”. And there are other hadiths which describe righteous deeds as being part of faith.” (End quote from Majmu` Al-Fatawa, 7/13-15)
Shaykh Ibn `Uthaymin (may Allah have mercy on him) said:
“If one of these words is mentioned in conjunction with the other, then Islam is to be understood as referring to outward submission, which includes words uttered and physical actions on the part of both the believer whose faith is strong and the one whose faith is weak. Allah, may He be Exalted, says (interpretation of the meaning):
{The bedouins say, `We have believed.’ Say, `You have not [yet] believed; but say [instead], “We have submitted,” for faith has not yet entered your hearts.’} [Al-Hujurat 49:14]
These words and actions may also come from a hypocrite; he is regarded as being a Muslim outwardly, but a disbeliever in his heart.
Iman is to be understood as referring to inward submission, which is affirmation in the heart and deeds of the heart, which only comes from one who is a true believer, as Allah, may He be Exalted, says (interpretation of the meaning):
{The believers are only those who, when Allah is mentioned, their hearts become fearful, and when His verses are recited to them, it increases them in faith ; and upon their Lord they rely – The ones who establish prayer, and from what We have provided them, they spend. Those are the believers, truly.} [Al-Anfal 8:2]
In this sense, faith is higher in status, so every believer is a Muslim, but the opposite is not true. (End quote from Majmu` Fatawa wa Rasa’il Ibn `Uthaymin, 4/92)
The verse quoted in the question confirms this meaning with regard to the description of the family of Loot (peace be upon him), as they are described once as believers and once as Muslims.
What is meant by Islam here is outward Islam, and what is meant by faith is true faith in the heart . When Allah, may He be Exalted, referred to the people of [Loot’s] family as a whole, He described them as being Muslims, because the wife of Loot (peace be upon him) was one of his family, and she was outwardly Muslim, but in fact she was a disbeliever. When Allah, may He be Exalted, referred to those of the family who were brought out and were saved, He described them as believers (interpretation of the meaning):
{So We brought out whoever was in the cities of the believers. And We found not within them other than a [single] house of Muslims.} [Adh-Dhariyat 51:35-36]
Shaykh Al-Islam Ibn Taymiyah (may Allah have mercy on him) said:
“The wife of Prophet Lut (peace be upon him) was a hypocrite and was inwardly a disbeliever, although she appeared outwardly to be a Muslim like her husband. Hence she was punished in the same manner as her people. This is the state of the hypocrites who were with the Prophet (blessings and peace of Allah be upon him); they submitted to him outwardly, but inwardly they were not believers.” (End quote from Jami` Al-Masa’il, 6/221)
He (may Allah have mercy on him) also said:
Some people thought that these two verses imply that the words Iman and Islam are one and the same, and they thought there was some contradiction.
But that is not the case; rather the second verse is in harmony with the first verse, because Allah tells us that He brought out whoever in the city was a believer, and He tells us that He did not find any except one family of Muslims, because the wife of Loot was among the members of that one family, but she was not among those who were brought out and were saved; rather she was among those who were left behind and were punished. Outwardly she followed her husband’s religion, but inwardly she adhered to her own people’s religion, and betrayed her husband, telling her people about his guests, as Allah, may He be Exalted, tells us about her (interpretation of the meaning):
{Allah presents an example of those who disbelieved: the wife of Noah and the wife of Lut. They were under two of Our righteous servants but betrayed them.} [At-Tahrim 66:10]
Their betrayal was in terms of religion, not in terms of marital infidelity. What is meant is that the wife of Prophet Lut was not a believer, and she was not among those who were brought out and were saved, so she is not included in the words (interpretation of the meaning): {So We brought out whoever was in the cities of the believers} [Adh-Dhariyat 51:35], although she was a member of the Muslim family of Lut, and she was in the house with them. Hence Allah said (interpretation of the meaning): {And We found not within them other than a [single] house of Muslims.}
Thus we see the wisdom of the Quran, as it describes those whom it says were brought out as being believers, and it describes those who it says were in the house as Muslims.” (End quote from Majmu` Al-Fatawa, 7/472-474)
Shaykh Ibn `Uthaymin (may Allah have mercy on him) said:
Allah, may He be Exalted, says in the story of Loot (interpretation of the meaning):
{So We brought out whoever was in the cities of the believers. And We found not within them other than a [single] house of Muslims.} [Adh-Dhariyat 51:35-36]
Thus He differentiated here between the believers and the Muslims, because the family that was in the city was outwardly a Muslim family, as it included the wife of Loot, who betrayed him by disbelieving, and she was a disbeliever. As for those who were brought out from the city and saved, they were true believers, for faith had entered their hearts.” End quote from Majmu` Fatawa wa Rasa’il Ibn `Uthaymin, 1/47-49)
And Allah knows best.