jpeg is smaller(compressed) in size, the algorithm is "lossy" and jpg (jpeg) does not support transparency; furthermore jpg compression introduces "artifacts" which degrade the appearance especially when the image is magnified. ["Lossy" means that original information in the image (prior to applying the compression algorithm) is lost - you cannot retrieve the original image.] jpg files have a headder in which EXIF information such as shutter speed camera type, geotagging... is stored
png is often larger in size (unless it is a very simple image with great expanses of a single color) and supports transparency and is lossless. ["Lossless" meand that the original information is retained and is retrieved when the image is displayed] The png format has no (standardized) way of including EXIF data - which adds information about the picture such as shutter speed camera type, geotagging...
Obviously you see the advantages of png but if file size is the most important and the image is complex then jpg often is good enough. Both are widly supported by Web Browsers and image processing software. Few cameras save in png format
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