How would you translate french "bœuf" to american english? Well, we have, "beef". But someone would also just use boeuf which might have a closer recemblance with the pronounciation of the original word. So whats most important, to have an as direct as possible spelling to how it is pronounced or just make an own version of it.

We have the same with a lot of names in the west as well, Jacob, Jakob, Yakob, Yakov, Jakub, Jakup etc.

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.

The spellings reflect different transliteration schemes that can be used for Arabic.