A complaint is the document which actually charges you with a criminal offense (at least until the grand jury moves to indict you--then the indictment becomes the charging document). It is the item that makes the other paperwork necessary. The police report is a set of documents which sets out investigatory findings, interviews, and basic procedures that were followed before and after charging you with a crime.
Any answers given here do not indicate the existence of an attorney-client relationship between you and me, or between you and my firm. In order for such a relationship to develop, I must be admitted to practice in your jurisdiction, you and I must countersign a retainer agreement, you must pay my fee, and we must speak in confidence. If all of these conditions are not met, I am essentially answering a hypothetical for the purposes of discourse.
A Complaint Report is one one of many types of police reports. Some jurisdicditions turn over the police reports much sooner than others. New York County has a policy where they turn over these documents as late as possible.
I'm not sure there is a difference -- it may be different names for the same thing. Generally, when someone gets arrested, the police generate a report that lists the crime, witnesses, etc. When the person gets to court, the person is giving a complaint that lists the charges and a few facts. This document is usually drafted by a prosecutor and signed by the officer.
I am a former federal and State prosecutor and now handle criminal defense and personal injury/civil rights cases. Feel free to check out my web site and contact me at (212) 385-8015 or via email at Eric@RothsteinLawNY.com. The above answer is for informational purposes only and not meant as legal advice.
A Complaint, whether a "police" Complaint, Misdemeanor Complaint, or any other document titled Complaint is the document used for the basis to commence a prosecution for the crime charged. It spells out the basis to substantiate the charge and then adds the specifics of the crime to support the charge.
A police report is an administrative document within the police department that spells out the details of the incident in the officers own words on what occurred. It is discoverable to the defense and may even contain some valuable information that the defense may be able to use. Police departments usually have a policy on writing reports, whether called an incident report, police report, etc. Some police officers don't even submit reports for the incidents they handle on their shift.
Also, the police officer may take notes in a notebook or piece of paper regarding the incident that occurred. These notes are considered "Brady material" and are also discoverable to the defense. If notes were taken and were not given to the defense before trial, it may be grounds for reversal of the verdict depending on what were in the notes.