As I understand it, rating is your ELO, what the queue looks at to try to find similarly rated players. Rank is based on your rating, but changes only once a day- rating should change after every ranked game you play.
Rank is the icon, rating is the score. Battlemaster rank make the rating visible in game, otherwise, the rating is only visible on the website via Search for Player.
ATM you can have a very high rank and a very bad rating due to how the new system work,
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