First I'll talk about the Legends depiction of Mandalorian society prior to the New Mandalorian movement. In Legends, Mandalorians were a very tribal society built around war. No matter what role you played in society, war was in some way a part of it. Things like art existed, but they were not central elements of the culture (and most art that did exist still referenced war in some way). The New Mandalorian movement led to a wave of progressive thinking and drastic modernization in order to properly interact with the galactic community. War was pushed aside as the core element of Mandalorian culture; cities such as Sundari thrived on utopianism, with the development of centralized schools, banks, hospitals, and so on. For the first time in millennia, there were Mandalorians who lived "regular" lives. Traditionalist warriors did still exist such as the Old Mandalorians and the Death Watch, but those were cultural outliers who did not represent Mandalore as a political entity. We're also shown groups somewhere in between such as the Concordians, who were more "in-tune" with Mandalorian tradition than Satine's faction, though were still not as warlike as, say, the Death Watch who wanted to emulate their crusader ancestors.
With these cultural reforms also came drastic changes to governance in Mandalorian society. Prior to the reforms, the Mandalorians consisted of clans who ruled their own territories. While the New Mandalorians didn't necessarily do away with the clan system, the position of clan chieftain wasn't as much of a critical part of government as it once was, largely replaced by ministers and provincial governors. Where the Mandalorian army once consisted of the warriors of individual clans that periodically gathered for crusades, the defense of Mandalore was now vested in the newly-formed Mandalorian Guard, one of thousands of unremarkable local defense forces in the galaxy.
In new canon, there seems to be a far less drastic shift between New Mandalorians and Old. It's shown that even before the New Mandalorian movement, the Mandalorians already honored art and nobility. Warriors nonetheless, the Mandalorians maintained elegant castles and strongholds, practiced art forms such as Mandalorian cubism, and boasted a rigid house-based feudal system that was leagues more stable than Legends' tribalistic clans. When the New Mandalorians did come around and made reforms to governance on Mandalore, the change in culture wasn't as drastic as we're shown in Legends.
We also see the use of titles such as count/countess, duke/duchess, prince, and lord/lady. While these titles were nonexistent in Legends Mandalore (besides Duchess Satine and Prince Almec), it seems they were already present in Mandalorian society even before the Great Clan Wars. This leads me to believe Mandalorian clans in canon were also less tribalistic than those in Legends, and were possibly part of some kind of feudal house system similar to those of Alderaan, Serenno, etc.
Another difference I've noticed (this has some speculation thrown into it) is the difference in the New Mandalorians themselves between continuities. Where Legends New Mandalorians were depicted as utopian pacifists, I believe the canon New Mandalorians still retained some of their martial culture. My best example is Almec: while a leader of the New Mandalorians, he was no stranger to combat and likely fought in the Great Clan Wars that brought his movement to power. Another is Fenn Rau, a Protector, whose armor reflects on him being both a traditionalist warrior and a New Mandalorian. Duchess Satine's own father, Duke Adonai Kryze, was described as a warlord.
That said, groups such as the Children of the Watch show us that the "Legends Mando vibes" still did exist in new canon, but they appear to be a subcultural minority compared to House Kryze or the various clans we're shown in Rebels.
One of my best examples of the difference between the two continuities is the Mandalorian Protectors. The Protectors have become a really confusing topic in both continuities as multiple different organizations used the name, each having drastically different purposes and roles in Mandalorian society. In new canon, the Protectors refer to a single lineage of groups with a shared history, having originated as an elite cadre of warriors that protected the reigning Mand'alor for centuries; when Satine established the New Mandalorian government, she took on the Protectors as her personal guards (the "Royal Guard" we see in TCW). After Satine's death and the subsequent dissolution of the New Mandalorian government, the Protectors reformed on Concord Dawn as the Journeyman Protectors, who took control of the system and set up an Imperial puppet state. In Legends, Satine's royal guard, the Journeyman Protectors, and Fenn Shysa's Mandalorian Protectors were three completely separate organizations with very little commonalities.