There's no much difference between them but "проживать" is used primarily when you talk about living somewhere for some period of time (I.e. Он проживал там в течение долгого времени). Жить is used in more common meaning (I.e. Он жил долго и счастливо) and you cannot replace it with проживал here, but you can always replace проживать by жить.

Sophie, in this specific sense it means live at a certain place/address. When you say "I live at ..... street' the word "live' has sligtly different meaning than in "I live well". It means that you are based at ..... street. (though of course you can understand it literally: "I live my life at ... street") Here 'проживать' can be used, e.g. official/bureaucratic "проживает по адресу". It is more or less synonymous to жить. It just emphasises this specific meaning, distinguishes it from simpler and more general 'live' - disambiguates, and adds nothing. Today it is mosly bookish language. Also it survives in the language of officials. No surprise, bureaucrats would not like simple and philosophical concepts as 'live' in their papers:) You can meet it in novels too, so it also has literary (non-official!) connotations. But in daily life, at least in Moscow... I can imagine my friend speaking this way of another friend either - when it is deliberate play with style. - to disambiguate something. People use synonyms for this sometimes. E.g. "at this moment she lives at ... street" - to emphasize the 'address'-sense and the fact that they are speakin 'literally'. ...but it is rare even this way. I suspect it could be more common in past: this would explain my 'bookish' associations with the word.

живет is the same thing like проживает. May be "проживает" is a little more officialy and is used in reports and some articles.