Transliteration schemes from one language/alphabet to another are not always consistent.
If I'd have to guess, Mehmood is what is used in Pakistan, reflecting local pronunciation, whereas Mahmoud is the more traditional Arabic translation of the name (which is Arabic).
The "oo" and "ou" spelling difference has more to do with the english vs french influence, with Pakistan having had more English than French influence.
If I'd have to guess, Mehmood is what is used in Pakistan, reflecting local pronunciation, whereas Mahmoud is the more traditional Arabic translation of the name (which is Arabic).