When you see the term "public liability" attached to a business, its products, or its services, it refers to a specific type of legal liability that could result from injury to a member of the public while on company premises. There is an entire subsector of the insurance industry dedicated to providing various forms of public liability protection. On the other hand, legal liability describes the obligation of one party to pay a debt owed to another party.
Legal liabilities are part of civil law and criminal law. For example, if a company commits fraudulent accounting practices or fails to comply with government labor laws, it may face charges in criminal court and will probably be forced to pay damages to concerned parties. This is a criminal legal liability.
It is also possible that a company fails to uphold its end of a contract with another party. Contract disputes are almost always civil issues, not criminal, but courts still award damages. To win damages in a civil or criminal lawsuit, the suing party must prove both the guilt and the legal liability of the alleged violator.
Public Liability vs. Professional Indemnity
Strictly speaking, everyone has a public responsibility to not cause direct physical harm to other people or their property. This is the type of damage that public liability insurance covers. For example, a general department store like The Home Depot would carry public liability insurance in case a customer injures himself while handling tools inside of the store.
This is different from professional indemnity coverage. Professional indemnity refers to the additional responsibility accepted by a person or company that carries out a professional duty. The purpose of professional indemnity insurance is to protect against alleged breaches of professional conduct.