What is the difference between Amitābha's Sukhāvatī & the Lotus Vault? What is the relation of sukhāvatī to 蓮華藏莊嚴世界海 ("the lotus's vault, the perfected lokadhātu, the ocean") attested to in the Buddhāvataṃsakanāmamahāyānasūtra T279 &amp; Āryasaddharmapuṇḍarīkanāmamahāyānasūtra T262, wherein arahantaḥ, according to the Mahāyāna as attested to in aforementioned scriptures, perfect Buddhahood?

Context:


  Inquiry: in the Arhat's past lives the causes and conditions for being
  subject to embodiment necessarily ought to have been eradicated, in
  light of this they dwell where to perfect buddhahood?
  
  Response: when attaining arhatship, the three realms' myriad outflows'
  causes and conditions are exhausted, there is no more birth again in
  the three realms. There is a pure buddha land, beyond the three
  realms, where not even the word affliction has a name, in this kingdom
  of the Buddha, they hear the Dharma Flower Sūtra [i.e. the Lotus
  Sūtra], with this they perfect Buddhahood. As in the Dharma Flower
  Sūtra's words: "There are arhantaḥ, for example, who’ve not heard the
  Dharma Flower Sūtra,themselves they call 'ones who have attained
  cessation'; I in another realm for them speak this matter, that you
  all shall become samyaksaṁbuddhāḥ."
  
  (Mahāprajñāpāramitopadeśaḥ T1509.714a9)


Furthermore, these pure lands, what relation do they have to ākaniṣṭaghanavyūha?
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.