A writ is just a written command given out by a legal authority like a court. It can be to enforce an action, or stopping an action from happening. Now, a Public Interest Litigation (PIL) is a form of writ, with just one specification that the matter is related to the general well being of the public instead of a particular litigant.

A petition is another type of writ, where the people raise a request to a legal authority demanding an action in a particular cause in the interest of general public. Usually, it’s signed by a number of people.

So, in short, PIL and petition are writs but the reverse is not always true.

I’ll quote a legal expert, Mr. M.V. Gupta, from one of the many law forums out there :

Writs are filed by individuals/corporates and other persons for reliefs in their own causes whereas the PILs are applications filed by any citizen for remedying the hardships faced by the public at large. PIL is not defined in any statute. It is the outcome of judicial activism to take cognisance of a cause at the instance of any person (whether he is personaslly affected or not) affecting the public at large. It is an exception to the doctrine of Locus Standi applicbale to actions in courts of law.

PIL
Public interest law loosely, refers to legal practices undertaken to help poor or marginalized people, or to effect change in social policies in the public interest, on ‘not for profit’ terms

WRITS
a form of written command in the name of a court or other legal authority to act, or abstain from acting, in a particular way

PETITION
a formal written request, typically one signed by many people, appealing to authority in respect of a particular cause

PIL is writ only but PIL means litigation in the interest of public and not in the interest of the litigant.
It is also important to understand basic difference between a regular Writ Petition and PIL Writ Petition.
Whenever a person affected by any illegal act or omission of Public Officials or of any Public office, he may approach the High Court for issue of appropriate Writ (authoritative direction).
However a person may approach the High Court for issue of appropriate Writ in the larger public interest even when he is directly not affected by illegal acts or omissions of Public Officials. I hope the difference is quite visible.
Can a Writ Petition be Treated as a Public Interest Litigation?
Yes, a writ petition filed by the aggrieved person, whether on behalf of group or together with group can be treated as a PUBLIC INTEREST LITIGATION however,
The writ petition should involve a question, which affects public at large or group of people, and not a single individual.
Only the effected /Aggrieved person can file a writ petition.
There should be a specific prayer, asking the court to direct the state Authorities to take note of the complaint /allegation.

Writs refers to constitutional remedy for all citizens under art 226. These can be in the nature of habeas corpus, mandamus, prohibitions, quo warranto and certiorari, or any of them, for the enforcement of any of the rights conferred by Part III and for any other purpose, where the complainant can directly approach the high court.

PIL stands for public interest litigation. Earlier it was only a person whose interest was directly affected along with others, whereby his fundamental right is affected who used to file such litigation. Now, the trend has changed, and, any Public-spirited person can file a case (PUBLIC INTEREST LITIGATION) on behalf of a group of person, whose rights are affected. It is not necessary, that person filing a case should have a direct interest in this PUBLIC INTEREST LITIGATION

Petition is a complaint filed by any aggrieved person for, award of remedy by court. Known also as ‘case’.

Writ: A formal, written order issued by the Court, which is to be obeyed by the individual/authority to whom it is addressed.

The Indian Constitution provides for writs for the enforcement of Fundamental Rights. High Courts can issue writs for purposes other than the Fundamental Rights, while the Supreme Court can only issue writs for the enforcement of Fundamental Rights.

Writs are of various types, such as Habeas Corpus, Mandamus, Certiorari, etc.

PIL (Public Interest Litigation): A writ filed by an individual in the interest of the public at large (rather than in the interest of the litigant, who might or might not have been affected). The main objective of a PIL is to protect public interest.

So, a PIL is a writ, but not all writs are PILs.
seeing relief on legal grounds. For example, a petition seeking issuance of a writ is known as a writ petition.