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Royall Tyler is a newer translation based on a "modernized" version of the original. Today the original is mainly read by scholars due to the 10th century language used. There are several "modernized" versions, ranging from post-war kana usage to a more modern narrative. A "new" version seems to pop up every few years or so, including manga versions. Your choice between Seidensticker and Tyler would kind of depend on whether you want to read it purely for relaxation/entertainment (Tyler) or to accrue knowledge of late Heian Japan (Seidensticker). If I recall correctly Genji was one of Seidensticker's earlier accomplishments and he kind of entered his pinnacle with translations when he did Tanizaki and Kawabata. Who knows, you might enjoy reading both.
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