A personal chef is essentially a professional chef who prepares meals for a customer or a family. This can happen in the customer's home or in the chef's own kitchen. A private chef, on the other hand, is a chef who works in a private home, usually for a single client or family. A personal chef is, in essence, a contractor. They cook for a customer base, often for several families, and are paid per day or work as a cook.
They don't work in commercial kitchens. A private chef works for a single family, an individual or a couple. Someone has hired them, paid them a salary and will most likely pay them benefits as well. They have fixed schedules or perhaps flexible schedules. The main difference between a personal chef and a private chef is that a private chef is usually employed by an individual or a family, while a personal chef is self-employed and has any number of clients on their list.
Possibly the biggest disadvantage of hiring a private chef is that for someone with an “average budget”, private chefs are very expensive. A private chef works full time for a client in their private residence, whether it's a house, hotel or yacht.