The reason old world monkeys and new world monkeys at first seem more similar is because they share ancestral traits.
The common ancestor of NWM and OWM and apes had a tail, walked on four legs, had 4 cusp teeth, and so on. But one type of OWM evolved a suite of new features. The reason NWM and OWM share things like 4 cusp teeth is that neither of those groups evolved the derived condition, instead they share the ancestral condition.
So in analogy, scaly skin is an ancestral condition in reptiles. One branch of the reptile tree evolved feathers. But there are plenty of reptiles with scaly skin who are more closely related to the feathered Reptilia than they are to other reptiles that also have scaly skin. So scaly skin isn’t a good diagnostic of relationships among reptiles. Shared ancestral conditions don’t indicate close relationships, shared derived characteristics do.
So suppose we have 4 creatures, 2 have scaly skin and 2 have feathers. We can say that the feathered creatures are probably more closely related to each other than they are to the other two. But since we know scales are ancestral, we don’t know whether the two scaly creatures are more closely related to each other than either is to the feathered creatures. And if the creatures are a sparrow, an ostrich, a crocodile, and a turtle, we find that the crocodile is more closely related to the sparrow and ostrich than it is to the turtle. Even though the crocodile and turtle share a lot of characteristics, it turns out that those are ancestral.