The answer is; possibly, yes. If the original file is a compressed image, and you take a screenshot of it and then save it to a lossy compressed image file type (like gif or jpeg), you would lose some of the fidelity compared to downloading the actual photo. If you save it with a lossless image file type (like bmp or png) then you would preserve quality but might possibly gain file size (the saved file would be larger than the original).
Additionally a photo might not be rendered at the original resolution as web pages and browsers can "resize" images, so the screenshot would be of the resized image, not the original. This can also cause image degradation if the image is stretched, or loss of resolution if the image is shrank.
If you happen to get the file at correct (i.e. whatever resolution you're subjectively expecting, not an objectively correct) resolution and without stretching or shrinking and save the file with a lossless format then you are getting the same image.
1And if the photo isn't much larger than your screen... – user1686 – 2019-04-10T06:49:59.720