What's the exact difference between 住 and 居?
住 http://www.cantonese.sheik.co.uk/dictionary/characters/668/
  
  [1] [v] live; reside; stay; dwell; inhabit [2] stop; cease


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  居
  
  http://www.cantonese.sheik.co.uk/dictionary/characters/1017/
  
  [1] [v] to live at, dwell, reside, inhabit, occupy, abode
  [2] [n] a residence, an house, dwelling 


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  居住
  
  http://www.cantonese.sheik.co.uk/dictionary/words/4174/
  
  to reside; to live (at some place) | [粵]: 住



住 is a verb for "to live (in); to reside (at) "
居 is mainly a noun for " a residence, an house, dwelling" 


Since the meaning of " to live / to reside " has a dedicated verb 住 , and 居 as a verb, has other meanings (for example: occupy) , few people would use 居 for the meaning of 住 , except in instances that literal style of reading is preferred. 

You can think of 居 is more literal and 住 is more colloquial when it comes to using a verb for "to live ; to reside"

Some example sentences:

他現在住在美國 (he currently resides in America)

他現居於美國 (he currently resides in America)

他現在在美國居住 (he currently resides in America)

*unlike 住 or 居, which is placed before the object , 居住 is placed after the object.