My child asked me, what's the difference between a Muslim and a Mu'min. I replied, Child ' It is like the difference between a Human being and being a Human. A human being is a general term for all who are born as a son of man. Whereas Being a human requires certain qualities. Likewise, a Muslim, in general, includes all those who have uttered the Islamic Creed, the Shahada. Then being a Muslim in practice is the Mu'min which requires certain qualities which Allah wants to have in a believer.
There are sure signs and actions which we are informed as that of the believer. Like a tree when roots are firm inside shows the fruits externally. A Muslim is Mumin when fruits of faith appear in his life. Almost all the promises of Allah in the Quran are based on the qualities of a Mumin.
We don't know what's in someone's heart. Hence we can only Judge someone to be a Muslim, and we cannot say or measure how much a Muslim is Mumin. For Example. A father who has two sons which one of them is a Hafiz, Alim and leads a pious life. Whereas the other is into a different lifestyle, who is hard to be recognised even a Muslim.
The father at his deathbed makes a will to give the larger share of his wealth to the one who is a pious child. Will the Islamic court agree on this?. No, both his sons will have an equal share of the inheritance as a son. Both are Muslim in the sight of the law. Hence in physical judgement in this life, all are Muslims. But in the sight of Allah, those who strive to have the qualities he wants to have in Muslim are the Mumin. Muslim is a label and being a Muslim (in truth) is a Mumin. To be a Mumin requires endurance and sacrifices to build the qualities of a believer that's within.
قَالَتِ ٱلۡأَعۡرَابُ ءَامَنَّاۖ قُل لَّمۡ تُؤۡمِنُواْ وَلَـٰكِن قُولُوٓاْ أَسۡلَمۡنَا وَلَمَّا يَدۡخُلِ ٱلۡإِيمَـٰنُ فِى قُلُوبِكُمۡۖ وَإِن تُطِيعُواْ ٱللَّهَ وَرَسُولَهُ ۥ لَا يَلِتۡكُم مِّنۡ أَعۡمَـٰلِكُمۡ شَيۡـًٔاۚ إِنَّ ٱللَّهَ غَفُورٌ۬ رَّحِيمٌ