Does wax protect my car?

Indeed, it does. Car waxes have been used to protect cars since the invention of the car.

Traditional car waxes are a combination of hard waxes, soft waxes and oils. Generally speaking, hard waxes make the product last longer, oils make the product shiny, and soft waxes make it easier to apply. Different products would be blended to give you the best of all worlds.  Some would be formulated for longer life or better shine.

Some products have additional ingredients to refract light and fillers to hide scratches. Others have abrasives so that they wax and polish at the same time. 

These days they are adding polymers and even ceramic to waxes and calling them hybrid waxes, or sealants.  Many of the manufacturers of these products have been in business for generations, and they know what they are doing, and so generally speaking, any name-brand product is going to be a good product.

However, the bottom line is that any wax-type product, retail or professional, is a sacrificial layer which will wear off, and burn off in the sun.  At best, they last a matter of months. This is fine for car enthusiasts who enjoy waxing their car, but it is getting to the point where they don't hold any advantages over ceramic and the latest polymer sealants, with even the latter providing exceptional shine. 

Waxes will always have their place, with wash-and-wax type products being especially useful, as these get topped up every time you wash your car.