Most of the time, either would be acceptable depending on the intended meaning of the sentence. However, there is a subtle difference. 'Like' implies comparison whereas 'such as' implies inclusion. In your first sentence, you're including Mexico City as an example of a busy city. In your second sentence, you are making a comparison between someone else and your brother. 'Such as' is not appropriate in this sentence ("your brother is a friendly guy, not such as my brother") and it doesn't read well. So there are scenarios where you would use one and not the other. Let me give you an example which highlights the difference between 'such as' and 'like'. If you say "I want to visit a historical city such as Florence or London" then you are including Florence and London as specific examples of places you want to visit. However, if you say "I want to visit a historical city like Florence or London" then you are comparing Florence and London to the types of city you want to visit but you don't necessarily want to visit Florence and London. Perhaps you want to visit Rome which has a comparable amount of history to Florence. However, most people would interpret this sentence as you saying you want to visit Florence and London. As I said before, the difference can be very subtle.

'Like' obviously has many meanings, but in this specific context of 'for example', they are very similar. As meraj said, 'such as' is a little more formal, but in general they can be used interchangeably. In fact, to a avoid repetition, we would probably use both if we were giving two lists close together in the same text. Your first example: - "Jerry lives in a busy, noisy and chaotic city such as Mexico city" doesn't really make sense though. He lives where he lives. It would need to be something like: "Jerry likes to live..." to sound right. Also '...is there a rule...'

This link is helpful, http://english.stackexchange.com/questions/129976/difference-between-such-as-and-like

Like is more informal than such as. Also, such as more appropriate to academic texts, because ' such as' is more precisely.