Well, wax is the classic method of protection for a car's paint. You really can't go wrong with a good wax. You should expect great shine, a glossy finish, good water beading properties, and durability between 1 week and 1 month. The biggest downside to me has been the durability of waxes.
Synthetic polymers are modern science's solution to old fashioned plant and animal wax. So, we can assume that more thought and effort went into custom making these new synthetic polishes, whether that means that the chemists succeed or not is another question. I've used some modern synthetic polishes that really sucked. Most noteworthy is Durashine, from the makers of Duralube, which also sucks. Yet many have had stellar results with synthetic products like Zaino, Klasse, and Blackfire. A well designed synthetic polish like these, should produce excellent finish, excellent water sheeting, excellent surface lubrication, and unbeatable durability. These particular products lasts for months, not weeks. I've had my Zaino protected car in the Palm Springs desert sun during windy 45mph wind and sandstorms for a week, pleasantly surprised that the layer of Zaino was not stripped away by the pelting desert sand. Some people would argue that the shine you get from a synthetic is not as good as what you would get from a good carnauba wax, and yes there is some truth to that.
It really is best just to try some of the stuff out there yourself and see what does the job for you. You'll find no shortage of opinions on what is the ultimate car care product.
Which is best?
Shine: toss up. maybe wax is slightly better.
Protection: I would say synthetic.
Durability: synthetic by far.
Cost: usually wax is cheaper.
Ease of application: this varies with each product.