Von Clausewitz is usually considered to be the great Western military theorist, but he's strictly modern, and he sticks to military theory. Sun Tzu's writings had a lot to say about politics and human nature. Given that, I'd wonder whether Machiavelli might be the better comparison.

I think the two big things that separate Eastern from Western military traditions is the use of massive, low-skilled conscript armies and the value placed on trickery, deception, and political maneuvering.

Maybe not quite the same thing, but Alfred Thayer Mahan's "The Influence of Sea Power upon History" basically influenced every admiral's naval decisions in WWI and WWII (both Eastern and Western) and caused some of the greatest naval battles in history. The only problem was that pretty much everyone around the world based their naval strategies on Mahan, so everyone was going in with these huge naval forces trying to destroy each other, causing massive battles. In fact, it continues to be extremely important and emerging countries are still using it as the basis of their naval strategy.

The west does not really have text quite as old and as well known as Sun Tzu. I am not a expert in how well known Sun Tzu was in the east.

Their are early text that can teach you about miltary matters but are not directly tactical or strategic guides. Thucydides "History of the Peloponnesian War" has lots of insideds into miltary matters and startegy. We have lots of text from Byzantian times, the Strategikon and the Tactica are well known. The Byzantian certently had a larger tradition of giving out manuals like this, their are a lot more. They often tried to combine roman and classical greek knowlage.

In the Renaissance the old Roman text were rediscovered and they informed leaders of the time and some ideas were taken over, others could not be used. These text were often not written by actual generals but rather historians and they may contain errors. This is less likly to happen in a dedicated book on the subject like the 'Strategikon'.

So at many times in history people were looking back on greek and or roman past.

In more modern times we have a explosion of text on these matters, Clausewitz beeing the most well known.

On a side note, what are some of the noticeable conflicts in ideas between the Eastern and Western approach to warfare?

Im am relativly sceptical about these 'cultural' differences between east and west. Their are however historians who belive in this, see The Western Way of War by Victor Davis Hanson.

The idea of the Golden Gate (leave an escape route open) was used by Western and Eastern Armies for different reasons at different times.

Decisive Battles were fought in the east and the west alike.

I can't say that it is as extensive as The Art of War, but De Re Militari by the late Roman writer Vegetius was popular and well-regarded during the Middle Ages. It did sort of fall out of favour, though, unlike The Art of War. I can sort of see why: I own a translation of De Re Militari, and while there is nothing in it that is "wrong" per se, it just really isn't that extensive or descriptive.

As others have mentioned, Carl von Clausewitz easily occupies a Sun Tzu-like position when talking about strictly modern history. A contemporary of Clausewitz, Antoine-Henri Jomini, also was very influential, though his fame hasn't been as lasting as that of Clausewitz.

how do we get to the assumption that eastern generals, or any generals for that matter, based their strategy on Sun Tzu? Sun Tzu's writings are a collection of interesting contrasts, "appear strong when weak" "appear weak when strong" etc etc. There's nothing there, that I can see to base any sort of military strategy on.

He makes observations on stuff, but it's not the same as suggesting how to win or what to do. Is there any evidence that anyone tried to put these ideas into practice?

Not a western reference, but still related to the topic : the 36 stratagems. A kind of eastern counterpart, and even probably better than the Sun Tzu, from my point of view.

https://en.wikipedia.org/wiki/Thirty-Six_Stratagems?wprov=sfia1