What is the difference between Amitābha's Sukhāvatī & the Lotus Vault?
Perhaps your title question is too board. And I'm too lazy to check all the Sanskrit names of the Sutras in corresponding to the Chinese.

So, you are asking to compare Sukhavati with the Lotus-treasure World? Let's give it few directions:


Sukhavati and Lotus-treasure are not in the same category. Category means when you can compare China with USA, but you can't compare China with America. Here USA and China are names for the countries, America is the continent. But you can compare America with Asia, both are continents. 
Lotus-treasure World includes Sukhavati. Why? Because Lotus-treasure World is a generic name given to all existences and phenomena, all non-existences and non-phenomena, i.e., the all of all, the total of totality. In the Avatamsaka, it said the Lotus-treasure World is formed by innumerable world-seas and world-seeds. Sukhavati will be one of the world in one of the world-seeds.   
Sukhavati is very far from Sahā-lokadhātu our world. One is it is mentioned in the Amitabha Sutra, said it is at certain and certain huge distance from our world of the western direction. The other is, in Avatamsaka the chapters describing the structure of the worlds, after it described the world-seed our world resides, it goes to describe the neighboring ones; when it depicted a world with known relationship to our world, say, both are at the 13th dimension, it stated clearly, also at the 13th dimension, such and such world is called so and so name... But it never mentioned the world Sukhavati. Therefore we can know Sukhavati is not a neighborhood world to ours.


From my reading, it seems Sukhavati is a "purified" world, that is, it is emanated from Buddha Amitabha's mind. But Lotus-treasure is the primordial, it is the manifestation of Vairocana.