What are the differences between the 1080p version and the 720p version of the same media?

Differences in video probably won’t be noticeable for untrained eye. 1080p video would have to be downscaled anyway.

Let’s assume the original video was 1080p. In this case the 720p video was first scaled, then compressed. On the other hand, 1080p clip was first compressed server-side, then scaled on your machine. 1080p file will obviously be bigger. (otherwise it would offer higher resolution, but at lower quality, ruining the visual experience and invalidating the point of using higher resolution1)

Lossy compression usually causes visual artifacts that appear as square blocks with noticeable edges when video is paused, but aren’t visible when you play it with normal framerate. 1080p file will contain more square blocks (caused by compression) than 720p video, but those blocks will be of approximately the same size in both videos.

Doing simple math we can calculate that 1080p video will contain 2,25 times more such blocks, so after scaling it down to 720p those blocks will be 2.25 times smaller than in actual 720p video. The smaller those blocks are, the better quality of the final video is, so 1080p video will look better than 720p video, even on 720p screen. Resized 1080p video will appear slightly sharper than actual 720 clip.

Things get a bit more complicated if source material was bigger than 1080p. The 1080p clip is first scaled to 1080p and compressed before you play it and then scaled once again while playing. The 720p clip is scaled only once and then compressed. The intermediate scaling step which is present in 1080p video case will make its quality slightly worse. The compression will make 720p even worse, though, so 1080p wins anyway.

Audio

It’s not only video that is compressed, but audio too. When people decide to use higher bitrate for video compression, they often do the same with audio. 1080p version of the same video may offer better sound quality than 720p video.

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