Identify important features of Cleanthes's and Paley's arguments and how the two differ from each other and articulate objections raised against each.

To begin, it's important to note that Cleanthes is not actually a real person but a character invented by David Hume in his Dialogues Concerning Natural Religion. In this work, Cleanthes is in conversation with Philo, a skeptic, and Demea, who adheres to a more Orthodox brand of Christianity. So, the objections and criticisms to Cleanthes's arguments are raised in the very same work that his argument is presented.
Both Cleanthes and Paley advance versions of an argument known as the Argument for Design. Cleanthes claims that the existence of a Creator is the rational conclusion that arises from study of the universe itself. He compares it all to a "great machine, subdivided into an infinite number of lesser machines, which again admit of subdivisions to a degree beyond what human faculties can trace and explain" (book 2). In Cleanthes's view, nature's mechanical quality compares closely with the mechanical creations of human beings, though nature achieves levels of sophistication no person can match. For Cleanthes, the conclusion is a matter of "the rules of analogy": if the "effects resemble each other," then it must be "that the causes also resemble" (book 2). Therefore, he concludes that there must be a Creator, and that Creator must possess rational intelligence.
Hume criticizes this argument in the very same Dialogues in which he introduces it. For one thing, the skeptic, Philo, argues against it on epistemological grounds, holding a more empiricist vision of knowledge, where knowledge must be founded in practice and experimentation. On these grounds alone, he rejects the argument. Additionally, he notes that the argument is weak in the way that it relies solely upon its use of analogy, and he points out that one could easily advance alternative analogies from which radically different visions of God would take shape. For example, Philo asks, can we not just as easily compare the universe to some kind of incredibly vast and complicated organism? In this case, we would need to start devising a very different vision of God.
Paley's Watchmaker Argument is, like Cleanthes's, at its core an analogy. Paley would have you imagine yourself stumbling upon a watch (and he notes that it would be irrational to assume that said watch had always existed). If you were to take apart the watch and switch around its internal mechanisms, the watch would no longer function. It has internal complexity, and this demands a designer. Paley argues that the same logic applies to the Creation, with all its variance and internal complexity (far beyond anything created by human hands): it likewise demands a Designer to explain it.
According to the Internet Encyclopedia of Philosophy, this version of the argument is an improvement over Hume's earlier statement of the Argument from Design, being that it "does not depend on . . . a general resemblance between the objects of comparison" (IEP). For Paley, all that really matters is that both the watch and nature share and exhibit that same quality of complexity, and this is what requires explanation.
As the Internet Encyclopedia of Philosophy proceeds to point out, however, the modern theory of evolution presents a scientific explanation for the complexity of nature without requiring the existence of a Creator. Rather, according to the theory of evolution, scientific forces have been shaping the Earth's biome over a span of millions of years, resulting in the world we see today, with all the complexity noted by Paley.
https://www.gutenberg.org/files/4583/4583-h/4583-h.htm