PNG vs PDF
PNG is an image format. PDF is a document format. Use PNG for a single visual image and PDF when pages should behave like a document.

Quick answer
Use PNG for screenshots, transparent graphics and image assets. Use PDF for multi-page documents, forms, submissions and files that should print consistently.
The main difference
A PNG is one raster image. A PDF can hold pages, text, images, forms, metadata and links. That makes PDF better for document workflows.
Examples
A screenshot of an app interface is naturally a PNG. A set of screenshots that must be submitted as one file is usually better as a PDF.
FAQ
Common questions
Can a PDF contain PNG images?
Yes. PNG files can be placed onto PDF pages, which is what PNG to PDF tools do.
Is PNG better quality than PDF?
They are different formats. PNG quality depends on image pixels; PDF quality depends on the content stored inside it.