I would like to know what the difference is between the two spellings of the
long o (ō), in hiragana, namely おう (ou) and おお (oo). Are these pronounced the
same?

If so, then why does Japanese have these two different spellings for the same
long o vowel?

Also, when transcribing from a textbook that uses romaji to teach Japanese,
and entering the characters at the keyboard, is there a set criterion for
curbing which combination I should be using (and dots it matter)?

Thanks.
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question
hiragana kana
Hiragana long ō: spelling: おう versus おお?
According to the wikipedia page, the difference is due to historical reasons.

お段の仮名の長音の場合には、おのかわりにうを添える  
例：こうしん（コーシン／更新）

お段の仮名でも、後続する音が歴史的仮名遣いで「ほ」または「を」であった場合には、「お」を添える  
例：おおきい（大きい）  
<https://ja.wikipedia.org/wiki/%E9%95%B7%E9%9F%B3>

Words that were originally written with を or ほ are now written with お.  
Example: おおきい (大きい)

All other words are now written with う  
Example: こうしん (更新)
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answer