There is no difference between the phrases. They have the same meaning, but sometimes people omit the "doing" at the end of the sentence.

I agree with Ben. 'How are you?' is the standard, polite formula in English for greeting someone and enquiring after their health or wellbeing. You can use this to anyone in any situation and it will always be the correct thing to say. 'How are you?' isn't a short form of anything - it's a complete question in itself. The answer is 'I'm well, thank you' or 'I'm fine, thank you', for example. You can use it to friends or to strangers, in either informal or formal contexts. This phrase will never cause offence and will never be seen as inappropriate. By contrast, 'How are you doing?' may be seen as over-familiar and even rude. It may not be acceptable in all situations. It would be inappropriate, for example, to say 'How are you doing?' to your boss at work, or to an older person whom you do not know well. As Ben says, this phrase is a 'riskier option'. It could easily cause offence in the wrong situation.

I would say "How are you?" is standard and safe. Learners can use it safely without causing offence. "How are you doing?" is perceived differently in different countries and among different social groups (in terms of informality, familiarity, seniority and all kinds of class and etiquette considerations) and is a riskier option for learners or visitors or outsiders.