Does it make a noticeable difference if I don't prepare the Turkish coffee in a Turkish coffee pot?

No, you don't strictly have to, although having a pot with a similar shape helps. For people who want to have a clean pot of the right size at all times, it makes sense to have a dedicated pot. Whether to use a traditional one or not is a matter of taste.

Nowadays, it is likely that you have a smallish pot in your kitchen which is clean (not soaked in old grease like the typical clay cooking pot in a medieval Ottoman kitchen), so there is no pressing need to dedicating a copper pot to the task.

What is convenient to have is a pot with one long grip, because you can pour the coffee through the strainer easier than when you have two small handles like on a soup pot. You also want the surface to not be too large. The coffee should be at least 2-3 cm deep, else too much water will boil out during preparation, and the coffee may overheat and scorch. If you are drinking small cups, you will need a smaller pot. The material is not important, as long as it is suitable for boiling water on stovetop.

But as long as the pot allows you to have your coffee deep enough, and you feel comfortable enough with pouring, any pot will do. My father has been using a smallish saucepan for decades and has always been happy with the results.