Hogwarts is protected by enchantments that disguise the castle as a crumbling ruin. All the other schools look on in contempt at this method - why pretend to be something lower than what they are?
Durmstrang is protected by its lake. That is, it is on the bottom of it. Muggles might scale the cliffs surrounding it, but none of them can ride their boat down beneath the depths. The castle waits for them, surrounded on all sides by the crushing darkness of the waters, one tiny refuge in world of madness.
The castle is protected by ancient magics, steeped in blood and sacrifice by desperate and angry men pushed out of their world by muggles armed not with wands but with flaming torches. The waters are held back by these spells but the darkness swirls constantly overhead – to warn the students of what might befall them should their magic ever cease to be anything but strong. Durmstrang students measure their fortitude based on how quickly they stop glancing fearfully overhead but instead learn to simply live with death hanging overhead like an old friend. In fact, many of them come to see the lake as such - an old friend. One who watches over them, protects them. But of course, only protects the worthy. The brightest and strongest.
(During the Tri-Wizard Tournament, Krum was not the only one to dive deep into the lake, craving the familiar depths below.)
Students might venture above, to fly their broomsticks over the snow-capped mountains or to sing songs in the cold crisp air as they tromp through their grounds. But always they venture back down to their refuge.
The first and most important lesson of Durmstrang is taught before their students ever enter its doors: the world is not kind, and will crush you if you let it. So you must be stronger, by any means you have.
Beauxbatons disdains Durmstrang’s blunt heavy-handedness, lifts their noses at the idea of doing anything as crude as sticking their marvelous school underneath a lake. How dreary! How cold and dark! Only Durmstrang students would lower themselves in such a way.
Instead, their glittering palace (palace, not castle) is protected by far subtler magics. They do not hide themselves away in mountains or in lakes, perish the thought! After all, what do you do out in the country? The Palace of Beauxbaton is located directly on the edge of Cannes, close enough to take the carriage into the film festival every year. Not to mention the shopping trips.
Their palace is quite famous in fact. Muggles come from all over the world to gawk at the beauty of their school, to stare in wonder at the glass and be bewitched by the fractured light. Beauxbaton students do not fear this. They pride themselves on it. For what better show of their power is it than to have the Muggles stare their magic right in the face and never see it?
The students sink through the glass, slide between the layers of magic and reality as easily as the light filters through their glass. Powerful magic to be sure, with every pane a different door and even the very floor itself a portal for the wizards who know where to step. To navigate their palace is a constant mental exercise – remembering which window leads to their arithmancy class and which leads to potions, learning to glide across the floor to avoid falling down into their Great Hall, resisting the disorientation that comes not from magic but from being surrounded on all sides by light and reflection. And of course making it all look effortless.
To call even the slightest bit of attention to the work they are doing would be alerting the Muggles that there is something that requires working at in what’s supposed to just be a glamorous art exhibit. They pay a price for the ability to giggle at the Muggles’ wonder.
But what they gain, oh that far makes up for it. Beautiful, subtle, powerful – those are the words that every Beauxbaton student learns to live by the moment they step inside their glittering palace. Magic means being able go anywhere, be anyone, do anything! They laugh in the face of Muggles and delight in their own brilliance.
Students graduate to be able to live anywhere they like, not crowded away in hidden villas like England’s Hogsmead or Rumpare-Malin Stan in Sweden. They are witches and wizards, not rats to scurry away to the safety of the dark. Above all, above even power, Beauxbaton learns pride.
They live in the world. They don’t hide away from it.
(Beauxbaton would also like you to stop comparing their brilliant, beautiful palace to some dingy old train station entrance, thank you very much Hogwarts)
(another by the incredible rainbowrites, considering these two schools not as pale and foreign imitations of Hogwarts but as wondrous seats of power in their own right, with their own fascinating logic.)