What's the difference between a TSO Authorization and a letter of TSO design approval?
I don't know much about this, but it looks like a TSO Authorization is for domestic (US) manufacturers, and a letter of TSO design approval is for foreign manufacturers who want to import their products into the US. See 21.621:


  §21.621   Issue of letters of TSO design approval: Import articles.
  
  (a) The FAA may issue a letter of TSO design approval for an article—
  
  (1) Designed and manufactured in a foreign country or jurisdiction
  subject to the export provisions of an agreement with the United
  States for the acceptance of these articles for import; and
  
  (2) For import into the United States if— [...]


And the FAA's page on it says:


  What is a letter of design approval?
  An FAA design approval for an article manufactured outside the United States that meets a
  specific TSO.


I have no idea about other aviation authorities, but I assume there are similar regulations in other countries because the same general issue exists everywhere: how do you ensure that both manufactured and imported aviation components meet all required standards?