It is important to realize that the differences between Griffons and Equestrians are not limited to biology or magical ability. The two groups possess very different views of how the world works.
To the Ponies of Equestria, the world is naturally peaceful and harmonious. If something or somewhere in the world is not in a state of peace and harmony, it is a result of outside tampering or some manner of corruptive influence. Discord is so horrifying to them (essentially an Eldritch abomination in the purest sense of the term) because in their eyes he was a violation of nature, an aberration that turned the world into a dangerous and chaotic place.
To get an idea of what this looks like in practice, one only need watch the very first episode of the show. The Mane Six describe the Everfree Forest, a place where the weather can't be controlled, plants grow on their own, and the animals take care of themselves, as unnatural.
Yet almost paradoxically, they possess a great respect and veneration for nature, albeit in what appears to be a very simplistic manner (trees good, polluting industry bad). When Twilight was traveling through time and witnessing all of the horrible futures that would have befallen Equestria had the Mane Six never met, one of them was one where the Flim Flam brothers had (apparently, we only got an eight second look at this particular future) constructed an economic empire. Twilight (and we, the viewers) witness heavy machinery being used to uproot and cut down whole forests, while in the distance, the smokestacks of factories fill the air with smog.
Consider that this timeline is shown alongside ones where Chrysalis and Nightmare Moon conquer Equestria. Implying that, in the eyes of Twilight, an Equestrian, that such an industrialized future is comparable to ones where her people are ruled over by tyrants.
The Griffons, by contrast, would look upon such an industrialized world with approval. This is because they see the world as dangerous and chaotic by default. Discord was not an aberration, he simply turned the unforgiving hostility of nature up to 11. To the Griffons of the Empire, the natural world is a thing to be combated, the cultural result of their homeland being filled with vicious predators that preyed upon them for countless generations. If something is peaceful or harmonious, it is only so because sentient beings made it that way through deliberate effort.
A bunch of Griffons leveling a forest to make way for a new rail line are not simply laborers working to earn their pay, or even industrious workers engaged in a beneficial act of construction, but soldiers in a war against a hostile, uncaring universe, destroying a potential haven for the monsters that used to prey upon them and their children.
Now, this is not to say that Griffons are a bunch of Captain Planet villains who will dump toxic waste into the ocean or choke the skies with smoke to "stick it to Mother Nature." In fact, the Empire has numerous laws that regulate how industries conduct themselves, from fishing quotas to a ban on the dumping of industrial byproducts into the nation's rivers. However, these laws only exist for the sake of the Griffons themselves. The fishing quotas are there to ensure they do not deplete their food supply, and they do not wish to drink from rivers tainted by toxic chemicals, or fly through skies choked by smoke. If there existed an action that would severely harm nature, but would be exclusively beneficial to sapient beings, Griffons would not only perform it, but feel a moral obligation to do so.
If it were announced that Manticores had gone extinct, Ponies would mourn the loss of an entire species, the death of a piece of the natural world that they could never get back. If any Griffons would be saddened, it would only be because they couldn't hunt the things anymore.
One must also remember that while Ponies are herbivores, Griffins are predators: half-lion, half-bird of prey. And their mindsets reflect this. Griffons tend to be quicker to consider violence as a potential solution to perceived problems, and tend to behave more individualistically than Ponies, with a higher rate of loners and introverts than the community oriented Equestrians.
With all of this in mind, it is perhaps unsurprising that Griffons possess a more martially inclined culture than Ponies. They are literally born with weapons built into their bodies, their talons sharp enough to sever an artery or tear through flesh. It is a common joke amongst soldiers and mercenaries that the only way to completely disarm a Griffon is to literally dis-arm them.
Hunting, a common pastime of Imperial Nobility and a means for citizens in rural areas to add extra food to their larders, is an almost foreign concept to most Equestrians. On occasion mobs of ponies or, in recent days, platoons of Equestrian Royal Guard, will sweep the wilderness to put down a manticore, cockatrice, or hydra that has been responsible for the deaths of their fellow equines, but the idea of doing such a thing as a form of recreation is completely alien to them.
Both Imperial Law and Equestrian Law hold murder to be among the worst crimes, but the courts of Gryphus tend to view acts of non-lethal physical violence more leniently than their Equestrian counterparts. Bar brawls and fistfights, not uncommon occurrences in the cities of the Empire, often aren't even prosecuted by the state if no citizens express the desire to press charges, which typically only happens if the fight in question resulted in a permanent, crippling injury. Griffons who press assault charges over a black eye are typically viewed with the same scorn as humans who sue a restaurant because a waiter spilled soup on them. To Griffons, theft is generally regarded as a greater crime than assault, while for the Ponies, the opposite is true.
And while neither Griffons nor Equestrians see war as something to be glorified or celebrated, with many Ponies seeing it as a last resort that should only be considered once all other options have been exhausted, only the Griffons see armed conflict as both inevitable and ultimately unavoidable. And despite, or perhaps because of so many Imperial families having lost loved ones to the fires of war, there does exist a certain romanticization of war and combat in Imperial Culture. Those who serve, be they knights, sailors or common soldiers, are adored and seen as heroes by the civilian population, and even the idea of dying in battle is looked upon with a certain degree of fondness. If the best way to die is of old age surrounded by loved ones, then the second best way is on the field of battle surrounded by your brothers and sisters in arms, giving your life for the nation and those you fight to protect, be they your family, your comrades or the Emperor.
While this is also true to a lesser extent in Equestria, they have not yet had this belief challenged by a massive, nation-spanning conflict like the Winter War. Any armed clashes they have witnessed were between individual Counts, Dukes and Barons in the days before the Alicorn Sisters unified the nation, with skirmishes smaller in scale and scope than the Battle of Wingbardy, codes of conduct between the warring factions, and casualty counts that never exceeded the low hundreds. Their understanding of war is thus quite limited.
The two nations even have differing ideas regarding the divide between the Nobility and the Common people.
The Griffon view of nobility, like so many other things, was irrevocably changed by the Winter War. At the Battle of Redstone, and the Peregrines, and the Storming of the Crystal City, nobles and commoners fought and bled side by side on the front lines. When the battles ended and the dead were accounted for, many nobles, officers and knights alike, were among them. They were cremated upon the same pyres as the lowest enlisted soldier or militiagryph, and when it came time to promote surviving non-commissioned officers to replace them, many commoners, battle hardened veterans, suddenly found themselves possessing equal or greater ranks in the military than members of the nobility. At first, this chafed at their sense of superiority...but as the war dragged on, and the promoted commoners proved themselves equal in skill and bravery time and again, in some cases saving the lives of their noble counterparts, the commoners earned their respect and admiration, if grudgingly.
This, combined with the settlement and establishment of the new province of Kestrella necessitating the elevation of numerous commoners to the ranks of the nobility to administer it (more than one of which was a Diamond Dog), the Post-Discordian economic boom leading to many merchants and bankers building up levels of wealth previously only held by Nobles, and rapid industrialization producing a burgeoning middle class of city-dwelling burghers, resulted in many of the social barriers between the Nobility and the common people being greatly weakened. There still exists a social divide, but it is much less than it was a few decades ago.
By contrast, the Equestrians, who have not experienced an equivalent war, sudden economic boom or rapid industrialization since the end of the Discordian Era, are still very much an agrarian, feudal society divided along class lines, with fewer opportunities for social mobility.
These are just some of the differences between the Empire and Equestria that are likely to result in confusion and culture clash at some point in the future.