On balance which are greater, the similarities or the differences?
There is a chance that your 'peasant' during the Carolingian period in southern France could have been a slave and not a free peasant, the odds depending on which historian's calculations you trust. By 1300 slavery had been extinguished in southern France, if not quite yet in the lands of their Spanish neighbours to the south.
By 1300 the wide gulf between Carolingian slavedom and free peasantry had narrowed and became, in custom and in law, a rather single entity: the peasant in servitude. But opportunity had also become more diverse with the development of technology and commerce. Southern France's position on the Mediterranean, with several important ports extant from antiquity (Marseille, Narbonne), had meant that import/export businesses never died off. But the south enjoyed a growth of towns perhaps faster than most other places in Europe in the 11-13th centuries. These towns were either brand new creations founded by either secular or ecclesiastical lords (bastide or salvetat, respectively) or based on Roman oppidium that had endured as villages. The rapid rise of towns was due to the huge growth of agricultural output and of commercial development it spurred.
These changes which spurred the growth of towns were founded on the lives of peasants, lives which changed in the 500 years between Charlemagne and the assertion of Capetian dominance over the French south. New lands were put to cultivation by more productive technology like 3 field rotation which made the earth more productive, horse collars which could pull faster and stronger, mouldboard plows which could turn heavier soil deeper, animal husbandry which resulted in greater animal variation to suit greater productivity.
Where the Carolingian peasant lived in earth and wood huts, 500 years later they could be living in stone houses; stone pulled from the fields by peasants' deeper, stronger plowing...or the cast offs of limestone mines for the castles and churches that sprang up across the south and which the peasants also built. It was a multitude of towns and castles and churches: some 200 new stone towns in the south sprang to life between 1100-1300. These towns drew in peasantry to work there in trades and industry, or to live behind the protection of walls and farm the fields beyond during the day.
So commerce and industry developed on the innovations of peasantry during this period such that the physical, agricultural world would have looked different over half a millennia.
But what of the spiritual world of the peasant?
The peasant of the Carolingian period, whether slave or free person, would have been Christian in name and identity. But in practice? They would have rarely been in a church or rarely seen a priest as we would know them, or as their descendants 500 years later would know them. By 1200 the Church was on a steady march into the souls of all the population, and perhaps no more so than the souls (and pockets) of the southern French peasantry.
The question of discipline of the average Christian in thought and action became paramount for the Church after 1000. Attendance to mass, sacraments like baptism and confession became obligatory and enforced by the 13th century. Marriage became subject to the Church. The entire life of the agricultural peasant in 1300 would have been enveloped by the Church (how effectively is another question).
And to add to the greatly expanded influence of the cathedral and monastery, by 1300 thousands upon thousands of potential heretic peasants of southern France would have been disciplined - interviewed, approved and/or corrected - by the inquisition: from Toulouse to Narbonne, Montpellier to the deepest Pyrenees.
Some reading:
On the transition of peasantry, Pierre Bonnassie, From Slavery to Feudalism in South Western Europe, (Cambridge University Press, 1991) and Georges Duby, The Early Growth of the European Economy: Warriors and Peasants from the Seventh to the Twelfth Century, (Cornell University Press, 1974).
On the 'world' of southern France in the middle of this period, Fredric Cheyette, Ermengard of Narbonne and the World of the Troubadours, (Cornell University Press, 2001).
On the inquisition, James Buchanan Given, Inquisition and Medieval Society: Power, Discipline, and Resistance in Languedoc, (Cornell University Press, 2001)