I wouldn't say that those are the two most popular sword designs, but they are both quite appealing. I don't consider most khuks swords, but some of them definitely have earned that title. If I see that blade shape on a long piece I tend to think "falcata" whether it actually is one or not. One thing worth considering is that the blade shape is not the only difference. For example, it's possible to make a much longer katana than khuk with the same amount of steel. In iron-poor Japan this meant that the katana was a much more practical design from a cost standpoint. A khuk is also more forwards-heavy.
Because they're made for different uses. Not only are their fighting styles very different but the khuk is suitable for a far greater range of tasks.
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The simple way to describe the difference is that a katana is better for saber-like draw cuts and stabbing. A khuk is better for chopping and violent shearing. It allows for a more natural forceful cut. Khuks are able to effectively bite into a greater variety of materials.