While the cosmological argument is based on the very existence of the universe, the argument from design (also called the complexity argument, or the physio-theological argument) is based on certain features of the universe or parts of it and the high level of complexity.
According to this argument, there are phenomena in the universe with such a high level of sophistication, chief among them being living beings in general and humans in particular. It is inconceivable that they were created by chance. The existence of these phenomena indicates that some intelligent factor planned and assembled them.
Suppose we were to walk in the desert and find on the sand not just a glass ball, but an elaborate and complex clock. Certainly, we would conclude that some intelligent creator designed it.
The cosmological argument is that the complex beings on Earth, especially humans, couldn’t have happened by chance. People that follow this stance believe there was intelligence or a creator behind it all. The primordial soup did not just happen; a chef put the ingredients together.