Praise be to Allah, and peace and blessings be upon the Messenger of Allah. To proceed:
When Islam and Iman (faith) are mentioned separately, each term encompasses the entire religion, so there is no distinction between them in this context.
However, when they are mentioned together in the same text, each has a specific meaning.
What Is Meant by Islam?
Islam refers to outward submission to Allah and includes outward acts such as:
Pronouncing the two testimonies of faith (Shahadah).
Establishing the prayer.
Giving Zakah.
Fasting Ramadan.
Performing Hajj for those who are able to do so.
What Is Meant by Iman?
Iman refers to inner belief and the actions of the heart. It includes belief in Allah, His angels, His Books, His messengers, the Last Day, and divine decree, along with love of Allah, fear of Him, hope in Him, and sincerity toward Him.
Which Is a Higher Level?
When the two terms are mentioned together, Iman is a higher level than Islam, because Iman relates to the inner state of the heart, while Islam relates to outward actions.
For this reason, scholars have said:
Every believer (Mu’min) is a Muslim, but not every Muslim is a believer with complete Iman.
How Do We Understand Allah’s Saying:
“So We brought out whoever was in the city among the believers. But We found there only one house of Muslims.”
Allah mentioned the believers when speaking about those who were saved because salvation was for those who possessed true faith.
As for describing the household of Lot as Muslims, this refers to the outward state of the household. Lot’s wife was among them and was outwardly a Muslim, but inwardly she was a disbeliever, so she was not among the believers who were saved.
Conclusion
When Islam or Iman is mentioned separately, each encompasses the entire religion. When they are mentioned together, Islam refers to outward actions, while Iman refers to the actions of the heart and inner belief. Therefore, Iman is a higher level: every believer is a Muslim, but not every Muslim is a believer with complete Iman.
And Allah knows best.