The phrases sound alike and the internet uses them interchangeably. They are three different things with three different sources of power, and picking the wrong one wastes time, or, in a dangerous situation, wastes time you don't have.
Who issues it: you, or a lawyer on your behalf. Who has to obey it: nobody, by itself. What it does: puts the recipient on notice, in writing and on a date you can prove, that specific conduct is unlawful and must stop, and that the evidence must be preserved. It is the first step in most civil disputes because it is fast, cheap, and often enough. Ignoring a well-founded attorney letter has a cost: it becomes proof in any later case that the recipient knew and continued. But the letter itself cannot fine anyone, compel anyone, or be "violated" in a legal sense. If you are searching for "cease and desist," this is almost certainly the one you mean, and the cease and desist hub walks through the three versions New Yorkers use most: defamation, harassment, and image misuse.
Who issues it: a court, or a government agency with the authority to regulate the conduct, the Attorney General, the Department of Financial Services, a licensing board, a federal regulator. Who has to obey it: the person or business it's directed at; violating it carries penalties set by the issuing body. What it does: commands a stop with the force of law behind it. You don't write one; you ask for one. In a private dispute that means a lawsuit and a motion for an injunction, which a court grants only on a showing that you are likely to win and will be irreparably harmed otherwise. For defamation specifically, New York courts are reluctant to order someone silent before the statements have been found false at trial, another reason the letter comes first.
Who issues it: a judge, in Family Court, Criminal Court, or Supreme Court. In New York the document is called an order of protection; "restraining order" is the everyday term. Who has to obey it: the person named in it, on pain of arrest. What it does: orders a person to stay away from you, your home, your workplace, your children, and to have no contact. It is the right tool, the only right tool, when there are threats, violence, stalking, or fear for your safety. It is not a tool for a business dispute, a bad review, or an ex who badmouths you to friends.
If you are in danger, do not send a letter. Call 911, and ask the court or a domestic-violence advocate about an order of protection. A cease and desist letter is for conduct that is wrongful but not dangerous.
Someone is saying false things about you, contacting you after you've said stop, or using your photo commercially: a cease and desist letter, first. A letter was ignored and the harm is serious and ongoing: a lawsuit seeking an injunction, a court order, which your letter now supports. A business is breaking a law a regulator enforces (deceptive practices, unlicensed activity): a complaint to that agency, which may issue its own order. You are afraid for your safety: an order of protection, today.
No. A restraining order, in New York, an order of protection, is issued by a judge and enforced by arrest. A cease and desist letter is private correspondence with no enforcement power of its own.
You can ask a court for an injunction, which is the civil equivalent. That requires filing a lawsuit and showing you are likely to succeed and will suffer irreparable harm without the order. Courts grant them sparingly, especially against speech.
Nothing automatic, there is no penalty for ignoring a letter. What changes is your position: you now have proof they were on notice, which strengthens a lawsuit and supports claims that their conduct was knowing.
Legally, no, neither is an order. Practically, yes: recipients and their lawyers read an attorney letter as the first page of a lawsuit, because it is, and respond accordingly.