Although it may seem intimidating at first, as it's quite different in English, it is actually quite simple once you understand this.
First of all, most native speakers alternate between the two tenses quite regularly, so in spoken French it's not too much of a problem. However, in formal settings, the futur simple is more likely to be used. (I.E. I WILL do something) The other main difference is the time scale of the futur simple. We would use the futur simple when referring to something that is going to happen 'actually' in the future: tomorrow, or next week for example.
The futur proche is just the opposite. (I.E. I am GOING TO do something). This has an informal feel, and is more common in spoken French. It is also used to describe something you are immediately going to do.
EXAMPLES:
Futur simple: je ferai un café demain.
Futur proche: je vais faire un café dans une minute.