If I take a photo by zooming in (optical) and cropping a photo without zoom. What would be the difference between the two?
I know cropping reduces the photo quality but I'm interested in other differences, if any.
Adjusting the focal length of your lens (ie, optical "zooming") will impact the depth of field of your image. This will change how much of the scene is in focus.
It will also subtly change aspect of the distortion of the image in order to project it on to a rectangular surface as lenses don't quite perfectly project their image and the exact variations change as the lenses are reconfigured for the different focal length.
It may also result in an increase in motion related blur indirectly as many consumer lenses may reduce their maximum aperture as you move towards the higher focal lengths. If you were shooting with the aperture wide open, this reduction in aperture may result in a longer shutter speed needing to be used to expose the scene.
Zooming will allow you to retain more information/detail in the photo. This would also leave you with the ability to then crop in on your zoomed in photo giving you an even closer "zoom".
Cropping is really just deleting unwanted part of your photo. i.e. you are taking a 12MP photo and only using 5MP of it, deleting the other pixels. Cropping should be used for minor composition adjustments.
however optically zooming would most likely lower your aperture (depending on the lens your using) requiring a longer shutter/slower shutter speed or higher ISO to maintain correct exposure.
There should be no difference apart from resolution if you crop from the exact center of the image... If you do not crop from the center then there may be noticeable distortion depending on how wide the lens is.
To prove this to yourself convert both images the crop and the zoomed version to the same resolution; then add them as separate layers in photoshop; finally turn the top layers blend mode to difference... the resulting image should be black indicating that they are identical.
Zoom first, and crop second. You can't crop what isn't in the original image, but the zoom can get you that detail. Also, if you are not using your zoom much, why carry it around? Better to use a fixed focal length lens and have much better image quality. It does blow my mind that you are not interested in image quality. With a photograph, what else is there? It's a visual image. Zooming does not always deteriorate image quality but cropping always does. When cropping one is always hoping that one is not cropping too much and destroying the picture!
Zooming in and cropping will enlarge the desired areas of the image, but so will getting physically closer to the subject. Taking pictures from too far away makes everything else more difficult. Of course there are times when you simply can't get closer, but getting closer is the way to go if you can do it.