For archival purposes, where you care about not losing details is more important than image size, PNG is better (though often TIFF is used for that use case). For images with large blocks of solid colors and sharp edges (text, line drawings), PNG is arguable better (though JPEG can be acceptable if you're careful with quality settings). If you need alpha support, go for PNG since JPEG doesn't support that.
For photograph-like images, where image size is important, JPEG is preferred over PNG.
For archival purposes, where you care about not losing details is more important than image size, PNG is better (though often TIFF is used for that use case). For images with large blocks of solid colors and sharp edges (text, line drawings), PNG is arguable better (though JPEG can be acceptable if you're careful with quality settings). If you need alpha support, go for PNG since JPEG doesn't support that.
For photograph-like images, where image size is important, JPEG is preferred over PNG.