It will be clear that French Christianity was going to be quite different, largely because the culture that was baptized with Clovis was so very different from the 'civilized' Romano-Gallic culture which had preceded it. At one level, the Arian Goths in the south and in Iberia were more 'civilized', to the point that when Clovis announced his intention to liberate the Catholic Gauls from their oppression under the heretical Goths, those same Gauls preferred to fight with the Goths against the Franks. They lost, though, and willy-nilly found themselves part of a new, Catholic, France.
The Romans thought that the Franks were rough, uncouth and bellicose. They had a point. It took nigh on a millennium to arrive at the court of the Sun King. There is little to be astonished at that Eastern Christianity looked with barely-concealed or even open contempt at the Western Church, so deep it was in the arms (in both senses) of barbarians.
But we mustn't let the East get away with it altogether; its history, too, was brutal, and Christianity did little to ameliorate that. But the East had both subtlety and scholarship, and it took the Franks a few hundred years to acquire these, beginning under Charlemagne who initiated a deliberate policy of recreating the Western Empire, and began it by himself learning how to read, and of importing all the finest scholars he could find (like Alcuin of York).
One curious difference between East and West was that the East was in a fairly constant state of war; first with the Persians, and then with the Moslems. The West was (comparatively) peaceful; it has sometimes been said (and I'm not really sure how much credence to give this) that, lacking a great deal of real all-out war, the Franks' friskiness was in due course to be channelled into the Crusades, rather than being wasted in beating each other up.
The creation of the Holy Roman Empire had another curious side-effect. Until this point, the Western Church had been fairly undisputedly governed by the Papacy. But in time the Holy Roman Emperors came to want the same sort of influence over the Church that their opposite numbers in Constantinople exercised. In the middle ages, it developed into almost a dual authority structure in the West; some Christians looking to the Emperor, others to the Pope, for ecclesiastical leadership. It was to cause the 'Investiture Crisis', concerning who had the right to appoint bishops. There were even different theological schools. Sometimes the Imperial Church (as it were) seemed in the ascendent, and sometimes the Papal, as when in 1076/7 the Emperor Henry IV was reduced to kneeling barefoot in the snow at Canossa to beg the Pope's forgiveness.
In some ways, this 'Imperial' church survived in France where (as in some other countries) the King appointed the bishops and governed the Church as every other aspect of the country. I think that perhaps you could make a case that this Imperial-style was what Henry VIII was aiming at in his break from Rome. Not a new Church, as such, but independent and secular government of the Church in England. To many a Frenchman, the 'alliance of throne and altar', which essentially dates back to Clovis, is of the very essence of France. France, in this view, is not truly herself without the twin pillars of the monarchy and the Church. They will not disagree, however, that it went very wrong. From the late middle ages, the state began to exploit the Church for its own ends. Abbacies (and of course their incomes) were conferred on laymen whom the King wished to favour. Consequently monastic life went into steep decline. Bishoprics were conferred on politicians and civil servants for the same reasons; and it did not seem to matter much whether the man was a good one or not. Talleyrand, for instance, was, though a known unbeliever, appointed to the bishopric of Autun in the fateful year of 1789 for political reasons. This effective, principled, autonomy of the French Church is what is known as Gallicanism.
There were other negative influences also. The wars of religion during the Renaissance were truly terrible. Catholicism triumphed in the end, but it waded through seas of blood in order to do so. The famous cynical comment of Henry of Navarre, a Protestant who converted to Catholicism when confronted with the resolute Catholicism of Paris which was the the only obstacle preventing his taking the crown as Henry IV, that 'Paris was worth a Mass' suggests that, really, the wars had done much to wipe out the fervour of the high middle ages.
Some fervour did return, of course. There were saints, great saints like St Vincent de Paul, St Margaret Mary Alacoque and many others, but much of the fervour was found in the new half-Protestant and 'charismatic' heresy of Jansenism. It, too, was ruthlessly stamped out, but, like Gallicanism, its ghost continued to haunt the French Church for many years. Some would say that it still does.