проживать 1. with a person's address/location, in the sense "to reside/dwell". Mostly in documents, normally we say "жить". In past it was more common. 2. imperfective from "прожить". Some thoughts about "imperfectivized" forms in general: Prefixes modify verbs' meanigns, but they also make verbs perfective (usually one is "just perfective", without change in meaning, like делать-сделать). "Переделать" means redo (once). When you need imperfective with the meaning of a prefixed form ("I'm redoing it") you complicate it more: "переделывать". We don't say "сделывать": here c- only adds usual perfective shades of completion/result/single-event. To use long "imperfectivized" forms, the shade added by the prefix must be important enough, and even then we sometimes use unprefixed form, just because it is shorter. "Прожить" is transitive. Я прочитал 5 страниц. Я прочитал книгу. Я прошел 5 километров. Similarly "a week" can be direct object of "прожить". All are about covering distances or spans of text, space and time. "Week" can also be adverbial, meaning "for/during a/the week": Две недели думал. Две недели жил в Москве. - "lived/stayed in Moscow for/during/in the course of two weeks". This one tells about your state ("living in Moscow") during two weeks. but: Две недели провёл в Москве. - "Spent two weeks..." Две недели прожил в Москве. - lived THEM in Moscow. Also "прождать два часа", "простоять два часа" (in a bus with no free seats, and you got tired). Many verbs can be used with "through-" and amounts of time. But "living" is a natural way of covering distances in time:) to be continued...

Проживать means "to reside, to be resident" and is way too formal and mostly used in official documents, e.g. "Гражданин X, проживает по адресу..." It also means "to spend a lifetime" but this meaning is used mostly in the perfective aspect: "Прожить жизнь зря/не зря/честно..."

It depends on a context. E.g. "To live in Moscow" (Я живу/проживаю в Москве) - both words have the same meaning. "I waste my live without any aim" (Я проживаю свою жизнь без какой-либо цели) - in this sentence russians never use word "жить". I can remember the only this one exemple to show the difference. In the most ways they have the same meaning.