Holy Qur'an:
"The Bedouins say: "We believe". Say: "You have not believed. However, say: 'We have become Muslims'. And faith has not yet entered your hearts. But, if you obey Allah and His Messenger, He will not decrease anything in reward for your deeds..." {Qur'an 49:14}

This ayah proves that while every believer is a Muslim, not every Muslim is a believer.

Shaykh al-Kulayni (d. 329H) records:

Muhammad b. Yahya - Ahmad b. Muhammad - al-Hasan b. Mahbub - Jamil b. Salih - Sama'ah:
I said to Abu 'Abd Allah, peace be upon him, "Tell me about Islam and Iman. Are they different things?" He replied, "Verily, iman takes Islam with it but Islam does not take iman with it."

I said, "Describe them both for me." He said, "Islam is the testimony that there is no god except Allah and belief in the Messenger of Allah, peace be upon him and his family. Through it, the blood is spared, and upon it arise women and inheritance, and upon its OUTWARD appearance are the majority of the people. As for iman, it is the guidance and what is established in the hearts from the characteristics of Islam, and what is apparent from action by it. Iman is higher than Islam by a rank. Verily, iman takes Islam with it on the apparent while Islam does not take iman with it in reality even if they both appear together in the statement and description (of a man)." {al-Usul min al-Kafi, vol. 2, p. 25, #1}

'Allamah al-Majlisi (d. 1111H) comments: Muwaththaq

Shaykh Hadi al-Najafi: The report has a reliable chain.

Again, Shayk Kulayni reports:
...Humran b. A'yan: I heard Abu Ja'far, peace be upon him, saying: "'Iman is what is firmly established in the heart, with which one comes to Allah the Almighty, and which is by action through obedience to Allah and submission to His Commands. As for Islam, it is what is outward from statements and actions, and upon is the majority of the people from all the sects; and through it the blood is spread and upon it arises inheritance and upon it marriage becomes permissible."
{al-Usul min al-Kafi, vol. 2, p. 26, #5}

'Al-Majlisi says: Hasan

Hadi al-Najafi: The report has a reliable chain.

They (Mumin & Muslims) are not equal indeed