Salam,
Firstly, we should keep in mind that when the Quran was being revealed, the primary audience consisted of
Arab pagans (idol worshipers), and
the jews and/or christians. First of them are called '
disbelievers', and the second '
people of the book', which means people to whom the previous divine books were sent to, and/or people who believe the veracity of previous divine books. Among the category of 'people of the book' were some who had forsaken the laws of God contained in their books and were deceived, And there were also among the people of the book who still believed in God and were submitted to him, and tried their best to uphold the laws of their scriptures. They needed time to examine the facts that Muhammad was indeed sent from the same God they believed, who had previously spoken to Noah, Abraham, Moses, Samuel, Joshua, David, Solomon, Jesus, and so on.
So, a muslim is one who submits to God and his beliefs and lifestyle are based in the laws given by Him through prophets in His scriptures. Noah's followers were muslims, Abraham's followers were muslims, Moses's followers were muslims, Jesus's followers were muslims, and so on. A follower of prophet Muhammad is also a muslim, but he is also named 'mumin'. A 'mumin' is a muslim who lived from the time of Muhammad and on. He is a muslim because he accepts and follows previous messengers, but he is a mumin because he has known and accepted the final divine message in its perfected, universal form, (which was not reached from a follower of Jesus for example). Note also the way the Quran addresses the Muhammad's followers in this verse: "The Prophet is closer to the believers (bil-mu'minīna) than their own selves, and his wives are their mothers."(Al-Ahzab, 6)(Yusuf Ali translation). Muhammad's wives are 'mothers of mumins', not mothers of every muslim since Noah's time, but only of the muslims from Muhammad's time and on.
Secondly, one may be a muslim but not yet a mumin. In what sense? well, one may have submitted to God and may show obedience with practice and purpose, but true 'imaan' (deep belief) has still not enetered deep in their hearts. For instance, he might have submitted to God, and his practices and purposes are towards God's pleasure, but yet when he is affected by a disgrace he's despaired and feels himself far away from God. Or he's still afraid from his enemies and he can't put his trust completely to God. You get the idea. It is ilustrated best in this verse: "The desert Arabs say, "We believe." Say,
"Ye have no faith; but ye (only)say, 'We have submitted our wills to Allah,' For not yet has Faith entered your hearts. But if ye obey Allah and His Messenger, He will not belittle aught of your deeds: for Allah is Oft-Forgiving, Most Merciful."(Al-Hujurat, 14)(Yusuf Ali translation).
Planting the imaan deep in our hearts, developing a steadfast belief and rely on God is not something easy, it requires patience, work with self-improvement, and studying the scriptures which help us know God better how he's like, what has he done for humanity in the past and what will he do in the future, because knowing these things about God is the basis of everything else - we can't love and serve a God we don't know. And this can't be achieved overnight. There are people who take these steps easier and in less time, and there are others who need more time and more effort. God knows us best (as he is the one who knows how he has created us). What grade of 'mumin' you reach and what depth has imaan entered your heart is something you need to work on. But the minimum He expects from us is that if we can't be steadfast mumin, at least be muslims (that is, to submit ourselves to him and practice as best as we can). This is why in this verse He uses the word 'muslim', not 'mumin':"O you who believe! Fear Allah as He should be feared, and die not except as muslims(muslimoon)" (Ali'Imran, 102). He expects from us that which is within our possibilities. Reaching the highest rank of 'mumin' can't be easy for everyone, but becoming a muslim is.
Thank God He made obligatory for me only that which I can surely acheieve, and if I can achieve more, that's even better! I am a muslim but I try my best everyday to become a mumin as well, so may God help me, and you brothers and sisters too.
I tried my best to be as correct as possible, and I hope I was of help,
Your sister Seraphina.
*Reference: "Muslim and Mu'min (believer) - The difference" by Joseph Islam