I feel like this is fairly good example of the different design trends.
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The skeuomorphic example isn't even that heavy, but you get the idea. Think the woodgrain and green felt from the original Game Center app. It's a literal visual translation of real word elements into digital ones.
Then flat design came, and honestly is solved a lot of problems that skeuomorphic design introduced. It allowed for more flexible designs that could be presented on a variety of screen sizes, resolutions, DPI, etc. Instead of using rasterized textures, it used vectors and code to draw the designs. Programmatic art/design essentially. But to its detriment, it lacked a lot of 'personality.' Design got really homogenous at this point.
Now neumorphic design is entering the mainstream. Design sites like Dribbble have featured work like this for a few years now, and big companies are starting to embrace it. A lot of the design/rendering is still programmatic (like with flat design), but now you can add more fidelity with things like gradients, lights/shadows that weren't as easy to support before.