What is the difference between an entertainment company and a production company?

These are not legal terms. If you see such a term in a law, it should also be defined in that law. For example, there are many laws that mention Adult Entertainment businesses, and those laws define what that term means. People call their companies what they want to call them. An entertainment company might be anything from a man who has a juggling and stilt-walking act, all the way to a multinational conglomerate that produces movies, TV shows, live shows, video games, and tours. A production company usually is going to be a company that is involved in some way in producing some sort of entertainment or journalism. That entertainment might be video games, live shows, TV shows, movies, or other forms of entertainment. The term "production company" is also often used as the entity under which an actor, writer, or director develops or produces movies. If you want to find out about a specific company, it is best to do good research on it.

A production company has some direct hand in creating the final product, such as developing a script, supervising hiring, getting the product shot, edited, and ready for market, etc. An entertainment company can be anything, run by anyone, and may deal with legal or illegal activities.