Skills that are only learned by professional chefs?
I don't think the skills are any different. The same basic skills apply and can be put to use at home or in a restaurant.  It's just that pros have a lot more practice, repetition, and refinement of those skills, and can thus execute at a  higher level, and consistently. Additionally, they have access to tools that might be too costly for most people to keep in their own kitchen...like a pacojet, or quality extrusion past maker...etc.  

They also accumulate more tacit knowledge accumulated from time in the industry, so that the skills can be employed in creative ways.  There are what you might call tips and tricks that are employed...something minor like placing a wet paper towel on top of herbs that have been prepped for a garnish comes to mind as a simple example...but I really can't think of anything out of reach for a curious home cook in terms of skills.  

However, the other thing to know is that pro chefs also must consider efficiency to a much higher degree than home cooks.  Not only efficiency in terms of speed and reduction of wasted steps, but also economic efficiency. I suppose many home cooks have to consider this as well.  

Maybe the bottom line is that pro chefs learn skills because they have to.  Home cooks can learn the same skills, often because they want to, but they aren't out of reach.