It actually matters a lot for a field called algebraic topology, which roughly speaking studies the properties of spaces and abstract surfaces, and especially knot theory. In algebraic topology you very often have to figure out when two surfaces or curves etc can be thought to be basically "the same" in certain senses and what properties that preserves. When you can smoothly deform one curve into the other, then the two curves can be thought to be similar in a sense and a lot of properties are preserved, so you can use some of the same tools to analyse them. The turning number is one of these preserved properties, and you can use it to characterize different curves. It's useful because instead of studying all the different and very complex curves and surfaces which crop up in various places, you can instead study a similar one that's simpler, and extend the result to all of them.

I think. I don't remember very well lol