The Towers, by my understanding, are known to basically maintain the reality of Nirn. After some reading, though, I got a bit of headcanon: what if they each have their own, unique effect on the world?
(I'll preface this by saying that I get all my information from wikis, 1d4chan and google; no direct sources)
This thought arose from Yokuda, the ancestral land of the Redguards and home of the Orichalc Tower. Back when Yokuda was around, the Redguards had a special magic/ability they practiced known as Sword-Singing. However, after Yokuda was destroyed, the ability faded. Two thousand years later, the greatest of it's practitioners could only manage the most basic form of it, and most recently there may not be anything left of it.
When Yokuda was destroyed, the Orichalc Tower presumably went with it. The Stone of the Tower was a sword. Hmm.
(have a bit of headcanon here that the Stone was specifically called the Left-Hand Sword, and the Redguard name for the local elves - the Left-Hand Elves - was derived from this)
The next thought was of the Throat of the World, otherwise known as the Snow-'Throat' Tower. The center of the one civilization that used the magic of Thu'um, or the Voice. Usage of the Voice started to fade after the defeat of Alduin, which also happened to be when the Tower was metaphysically damaged by the Time Wound. The last remaining practitioners of the Thu'um just so happen to spend their lives practicing near the top of Snow-Throat Tower.
The idea falters a bit here, as the Thu'um is also the magic of the dragons, and I rather doubt they are even remotely dependent on the Tower for their magic. Still, there's a certain theme. (side note: the Stone for Snow-Throat is a cave. No especially clear connection to the Thu'um, though it's possible said cave has very good acoustics)
Next one up diverges from this a bit - the White-Gold Tower. When the White-Gold Tower was deactivated (with the death of the Emperor), the barrier against Oblivion weakened to the point where Daedra could invade en mass. From what I've heard, it's believed that this is a side effect of a lot of Towers getting knocked down. However, the Khajiit Tower (if its a thing, which by my understanding it is) was disabled after that event, and no Daedra. Also the Crystal-Like-Claw was supposedly destroyed afterwords as well. It could be, then, that the White-Gold Tower is specifically geared for holding that barrier. But more pertinently, the Dragonfires - which are lit only while the Tower is active - are known to be a ward against Oblivion and prevent it's inhabitants from permanently existing in Mundus, and this has been the case for... a very long time, anyway. Longer than Red Mountain has been troubled. It's Stone was the Amulet of Kings. Which was clearly destroyed, and the Tower is clearly currently active, so what the Stone is now is entirely unknown. Moving on, about that Red Mountain...
Red Mountain. At first glance, it may appear that no special power resides in it's vicinity. Just a bunch of weird monsters, some of which are unaging due to a weird plague. And the Almsivi, godlike beings who draw their from... hmm.
The Corprus disease is a nasty thing. causing magic cancer and insanity. Also gives immunity to age and all other diseases. Was probably created by Dagoth Ur, probably utilizing the Heart of Lorkhan.
Dagoth Ur himself has ridiculous power, mostly or entirely stemming from the Heart of Lorkhan
The Almsivi are three Dunmer who gained godlike power, again from the Heart of Lorkhan.
The Heart of Lorkhan is the Stone of Red Mountain. One of Lorkhan's most notable traits is encouraging a sort of divine ascension for mortal beings (see: CHIM).
What if, then, all of this power does not come from the Heart of the dead god, specifically. What if this is the power of the Red Tower, drawn via the Heart?
Miscellaneous notes: there are several other creatures related to Dagoth Ur and/or Red Mountain, such as Ascended Sleepers and the Ash Spawn of Solstheim. What all such creatures have in common is 1: they were humanoids and 2: they gained powerful abilities from their transformation, the most common of which is a state of unaging.
It's worth noting that the Ash Spawn were created after the supposed deactivation of the Red Tower. However, what exactly happened is that the Nerevarine dismantled Kagrenac's enchantments on the Heart, and the Heart disappeared. There's no solid indicator that this is enough to deactivate the Tower. My headcanon here is that the Tower was destabilized, but is still active.
I'll note at this point that if the Orichalc Tower was truly destroyed with Yokuda, then it's power took a long time to fade, in which case effects may not necessarily be immediate.
It gets a lot fuzzier after this point. Running much more into headcanon here.
The Ada-Mantia is very unclear, but could be the primary, or even only Tower that maintains Nirn's timestream. Or not. Can't think of anything else it would do... although the effect Medora Direnni had on it was unusual. Maybe related? It's Stone is an event - the beginning of time and/or the Impossipoint Convention.
The Crystal-Like-Claw, sitting in the middle of High Elf territory, is generally assumed to be destroyed, as has been reported to have happened (destroyed during the Oblivion Crisis by Daedra, specifically). And what could it have done? Well...
There is a supposedly extinct order of mages that is based on a now-missing island near this believed-to-be inactive Tower that wield powerful and mysterious magics. (the Psijic Order, known to be active via the events of the Mages Guild in Skyrim). It's Stone is 'a person'. Or was, maybe. What the Psijics' power might have to do with Crystals, Claws or a particular person, I don't know.
The Green-Sap Tower could simply be powering itself. To elaborate, physical structure of Tree-Sap is a host of titanic, migrating trees that sometimes disappear for a while, only to reappear later. The power of Green-Sap could be it's own giant, mobile existence. On that point, it's worth noting that the trees have settled down. There's probably something wrong with it - metaphysical damage, maybe. It's Stone is a fruit, and latest information is that it's undamaged.
The Khajiit Tower is the Khajiit. It's Stone is their racial leader, the Mane. Who was killed, probably by Thalmor. There are two prominent magical effects that may be associated with this Tower. The first is the many breeds of the Khajiit, from the familiar humanoids to talking housecats to lions and whatever else. The second is the third moon.
The Walk-Brass Tower's power might just be what it has demonstrated. It breaks things, particularly time. So it's power could be breaking things, breaking time specifically, or maybe something more esoteric that is specifically related to time and the breaking things just stems from it being a absurdly huge golem. It's Stone is the Mantella and was the Heart of Lorkhan.
Now...
The Sload are a race of very evil slugs. A long time ago, the center of their civilization housed a structure called the Coral Tower. It was knocked down after the Sload (were presumed to have) unleashed an infamous plague that wiped out half of Tamriel and everyone on the continent joined together to salt some slugs. There isn't even the slightest indication (that I'm aware of) of what any Stone might have been. But for powers? Well...
Magically, the Slann are known for one thing above all else: necromancy. It also happens that after the Tower was knocked down, a giant whirlpool formed near where it's base was, perhaps as unfathomably deep as the Coral Tower was incredibly tall, and it is similarly quite mystical. Perhaps the Tower still exists in a sort of undead state? Also worth noting is there are no mentions of necromancy before the destruction of the Coral Tower (earliest canon on Necromancy was with Mannimarco, who came later. Mannimarco did not invent necromancy though; it's simply that nothing is known of it before him).
Biggest problem here is that, unlike all the other Tower powers, necromancy is quite widespread, well known, and easily practiced. To the point where it really doesn't fit in. Well, I'm not trying to be particularly rigorous with this anyways.
And a very headcanon bonus round.
The Tower of Atmora: Atmora is the old, abandoned home of the Nords, once supposedly temperate, now a frozen and lifeless wasteland. The headcanon is that the climate was sustained by a Tower, which channeled the cold into the magical ice known as Stalhrim. However, during one of the many Nordic wars, the tower was severely damaged, with inevitable consequences. The continent began to freeze, and stalhrim became increasingly rare.
(with this idea, Atmora isn't actually as cold as it could be, and restoring the Tower would both make Atmora habitable again and make Stalhrim much more abundant. Alternatively, the Tower has nothing to do with Stalhrim and was totally destroyed/deactivated).
This also gives an entirely new motivation for the Thalmor to not take the Imperial City. This is because they're trying to destroy the Towers. But if they destroy the White-Gold, they'll be fighting Daedra every step of the way toward destroying every other Tower. And if they capture they city, some brash minion might get it into his head to try to destroy the White-Gold, even without orders. So the best way to keep it active while they try to get the other towers is to not take the city.