"The shorter the focal length, the closer to the face. That is why the nose looks bigger in the 20mm lens shot than in the 200mm lens shot. If all photos were taken at the same camera distance, the face would look the same for each lens, however, the head size would change – smaller for the 20mm lens gradually getting larger up to the 200mm lens."

Shoot a bride from the floor, shoot a sports person from the tip of the hand or feet, shoot an ant eating a dog, shoot some toys next to real people.

Here's an example from Outdoor Photographer, photography (c) by Rob Sheppard. Note how the features of the tree in the foreground are highlighted, but are also set in the context of the mountains in the background:

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You would probably wish your lens could focus a bit closer than 20 cm if you want to make a picture like this: https://flic.kr/p/3Paa94

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Doing a close-up allows you to use perspective to further exaggerate the relative sizes. The wide angle also allows you to include more context than a telephoto lens, especially at close range to the subject.

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Dan Vojtěch's article: "Amazing how focal length affect shape of the face" demonstrates how changing your focal length and distance affects the appearance of everything. Pascal Vandecasteele uses the same photo in his article: "How focal length affect the shape of the face".

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"Wide angles create intimate relationships. Wide angles emphasize the photographer's height, distance, and relationship to the subject. When we look at a wide-angle portrait, the visual cues and exaggerated foreground-background perspectives tell us exactly where the photographer was, relative to the subject, within a broad environment.".

A very good way of using an ultra wide lens is to get really close and get something right in the viewers face. Using a wide angle lens to get everything in the frame often leads to very boring pictures because the viewer has no idea what the photo is about.

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This article from Photography life 'focuses' mostly on landscape photography with wide angle lenses: https://photographylife.com/how-to-use-wide-angle-lenses

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The reason I chose it is because it is very clear the creative possibilities of having a humungous first plane and a tiny far plane of the same person... Perspective.

Digital Photo mentor also wrote an article about how to use wide angle lenses: https://www.digitalphotomentor.com/5-mistakes-beginners-make-using-a-wide-angle-lens-and-how-to-avoid-them/

To me, the photos with the zoom lens appear more interesting for this purpose. The bokeh is stronger (even though all photos were taken at F/8.0) and the object fills more of the frame. What possibilities does an ultra-wide lens offer when I focus on an object that is close by?

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To get a close-up of something with a wide angle lens, you need to get extremely close. Everything seems about 3 times further away in the picture due to the focal length, so you need to get much closer than 50 cm to get something really close.

In the example below I took different photos of the same object, first twice with my Samyang 12 mm prime lense, then three times with my Sony E 18-200mm zoom lens, at various object and focal distances. I downsampled the images from 4000 to 1200 pixels wide but otherwise did not do any other post-processing than in-camera processing. The candle holder is about 9 cm high and 12 cm wide.

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At short focal lengths you are looking around the object and obtaining an in-focus background and with long focal lengths you are looking at a smaller segment from a distance, flattening the object and blurring the background. Of course it's possible to make adjustments to distances to increase or reduce the effects and crop a higher resolution to simulate a closer shot.

As @Agent_L says in the comments, if you take photos with your 2 lenses at exactly the same short distance from something, the answer to your question becomes more obvious. If you are curious about this - and you must be, since you asked here - then you should simply take the photos and see for yourself.

Ken Rockwell has written a very interesting article about his: https://kenrockwell.com/tech/how-to-use-ultra-wide-lenses.htm

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Different lenses are useful in different situations. Next time you are out in a forest or field, and come across an enormous mushroom, take a photo from 20cm distance using your 2 lenses, see the effect of the result, and the reason why someone might choose a close-up, wide-angle shot becomes self-evident.

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Probably the best comment comes from Em Jo Photo on DPReview's webpage: "Facial distortion of various focal lengths for headshots":

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For a longer explanation (and female models) check out JP's Slanted Lens video: "How Lens Focal Length Shapes the Face & Controls Perspective: A Lighting Tutorial".

The distortion of wide angle lenses allows you to magnify the subject relative to the rest of the frame, leading to dramatic or otherwise interesting compositions.

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I have modified the animation offered on Dan and Pascal's pages to slow it down, and repeated the first and last frames at an even slower speed to demonstrate the difference:

I have a Samyang F/2.0 12 mm for the Sony E Mount, so the 35-mm equivalent focal length is 18 mm (crop sensor 1.5×). I use it mainly for landscapes and architecture in order to fit more in my frame than I can with my other lens, usually focussing close to infinity (I also use it for photography of starry nights such as the milky way or star trails, as it is also by far my fastest lens). I sometimes use it in small indoor spaces, with a focus perhaps down to 1–2 metre at the lowest. But I've never used the lower half of the focus ring, which includes focus down to 20 cm. When I want detail, I tend to use my zoomlens (18–200 mm). Under what circumstances would one use a wide angle lens with a focus less than (say) 50 cm?