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.

First I'm going to answer the question you asked, then I'm going to answer the question I think you meant to ask.

Skills that are only learned by professional chefs

To start, "professional chef" is redundant. There is no such thing as an amatuer or home chef - chef just means "boss". The chef is the kitchen manager and in general is rarely working the line, if at all. A non-exhaustive list of things I can think of off the top of my head:

• Ordering chefs need to be able to review what they have available and accurately estimate the amount of business they're going to be doing over the next few days and how much prep they're going to need to do ahead of time. This also requires knowing what you can prep ahead of time, and how far ahead you can reasonably prep it, who delivers on which days (and when they don't), which purveyors deliver which products and the relative expected quality, and a whole host of other minutia I can expand on if desired.
• Shipping and receiving directly related to the above, you need to be able to compare a placed order with a delivery sheet with a physical delivery. You need to know which products are temperature sensitive, and how to check. Being able to eyeball weights and if you're getting the amount of goods you're being charged for, and how to deal with it if you're not.
• Menu writing what's in season? What can you actually get? How creative are you willing to be with ingredient descriptions? What the difference between haricort vert and green beans (not a single thing)? How much more can you charge for something if you gratuitously add truffle oil and/or lobster?
• Scheduling, conflict resolution, general worker interactions/discipline, etc As stated, the chef is the kitchen manager, and as such, is typically responsible for managing prep and line cooks, dishwashers, and the like. These would fall under the umbrella of "general skills you need to supervise/manage workers" and aren't unique to a kitchen (maybe with a few wrinkles due to language barriers/high stress environments/workers who carry a collection of knives as part of their work gear but nothing really out of the ordinary)
• Languages a basic understanding of Spanish and French while not a requirement, is extremely useful

I could go on but I suspect the question you're actually interested in is what skills are only learned by professional cooks. Following is another non-exhaustive, top-of-my-head list that I'll expand as it comes to me:

• Knife skills. I can already hear the home cooks screaming about this, and I'm sorry to say, no matter how good you think your knife skills are, they're not as good as you think. My wife is used to it by now, but I'll be cutting something up, look up from the cutting board to talk to someone without stopping, and not realize I've even done it until the other person freaks out that I'm slicing without watching. It's not just speed, but also accuracy and safety (a sharp knife is a safe knife and for the love of Escoffier please curl your fingers). There's also knife care, proper honing and sharpening, knowing which knife to use for which task (and how to deal with it when you don't have the knife you should be using). How to cut things properly in order to maximize their shelf life, various terms (julienne, brunoise, chiffonade, shudder tourne, etc), how to cut things without bruising them, how to cut things with bruising them, how to break down a bird, how to break down a rabbit, how to break down a fish, how to break down a flat fish, how to trim silverskin and fat caps off a steak, how to cut meat into x oz portions accurately.
• Food safety. Proper handwashing. Proper glove use. Avoiding cross contamination. How to organize a speed rack that has cooked chicken, raw fish, and prepped vegetables on it. Which color cutting board to use for which task. Why you can't use a service board for prep. The proper concentration of chemicals in your sanitizer bucket and how to check. The difference between a prep sink and a hand sink. What the minimum required internal temperature is for any type of meat, how long it has to stay there, and which rules you can ignore. How to store cold food on the line, how to store hot food, how to check the temperature, and how long you can keep it there. How to calibrate a probe thermometer. Major allergens/sensitivities and what they mean (a peanut allergy and a nut allergy are not the same thing), and which of your menu items have what in them. How to clean for an order for someone with a food allergy.
• Temperature estimation being able to tell what temp a burger is at by looking at it. Being able to tell what temp a steak or piece of fish is at by touch.
• Mental timer I can't really explain this one, but it's absolutely a thing that happens - I haven't cooked professionally for years but there's still a voice in my head that speaks up just before something's done or needs to be checked
• Recovering when something goes wrong what can you save and how can you save it? The breakfast buffet is empty, there's a line, an entire hotel pan of scrambled eggs is overcooked, how do you save it? You're plating a 4 top and drop a grilled porkchop in the sanitizer bucket, what do you do (no I never did that, why do you ask)?
• Physical pain tolerance it's a common sight to see cooks taking things out of a 500 F oven with bare hands. Any cut or burn that doesn't require a trip to the hospital you need to be able to bandage quickly and get back to work (for that matter, you also need to know how to bandage something properly in the kitchen and what's acceptable and what isn't). I've personally seen cuts cauterized in the kitchen with a blowtorch. Working in 100+ F ambient temperature for an entire shift - or organizing the 35 F walk in (or the 0 F walk in freezer) for an hour
• Various other miscellaneous speaking of six hours in a 100+ degree kitchen, corn starch is basically unmedicated baby powder and can be found in almost any industrial kitchen (do NOT get confused and use flour, it is NOT the same thing). I'm told the air circulator in the walk in freezer easily disposes of smoke with no lingering odors. Some wine from industrial food purveyors is absolutely disgusting but some of it is rather palatable - port is usually the strongest.

I think the real question you're asking is which skills professional chefs are taught are worthwhile to learn as a home chef?

Professionals have to do all kinds of things that you'd almost never do at home such as cook for 50-100 people every night, maintain huge quantities of stock, soup, ingredients, partially prepared/prepped/cooked food that can be finished to order quickly, etc. Basically be able to serve anything from a rather lengthly menu in 10-20 minutes. This is completely different than the typical home chef that only cooks 5-10 times per week and can plan in advance what you're going to have.

Also a lot of restaurant food is really intended to be extremely flavorful, "special", suitable for a night out, etc. which often means high in calories, salt, sauce, etc. This is great if you're going out to a restaurant but you would get sick of eating like this every meal, talk to people who travel for a living and their number one complaint is the food. Often you just want relatively simple and fresh food and just the quantities you want, rather than a big fancy restaurant presentation. So a lot of restaurant preparations don't translate too well to the home chef.

I think the best skills a home chef can adopt from a professional is quick and efficient preparation, being able to multi-task and time different dishes so they will all be ready at the same time, and learning how to change and adopt recipes based on what you have on hand rather than needing to make a special trip to the store for every meal.

This is why I am a fan of chefs like Jacques Pepin who have been professionals for a long time but can demonstrate how to cook a 2-4 course meal for 2-4 people in about 30 minutes based on ingredients that most people have on hand. This is the best and most practical, not to mention healthy, cooking that home chefs can do IMHO.