Linnaean taxonomy classifies apes and monkeys as two closely related groups. This is the classification system most people are taught in grade school.

Cladistics is a style of classification that seeks to organize species and groups of species from when they branched off of other groups of species. In this style, everything is defined by novel features, but they are still members of the more ancient clade. Birds for instance, would be a novel clade emerging from Dinosaurs, and thus all birds are also dinosaurs, but not all dinosaurs are birds.

Because there are two groups of monkeys with unique characteristics (new world and old world), and apes have unique adaptations not found in either group, we have no way of cladistically defining a monkey in a way that meaningfully does not also include apes.

As a side note, this is where the phrase "there is no such thing as a fish" comes from. 'Fish' in the Linnaean sense are a huge and diverse category. Two random members of the fish class would likely be far, far more distantly related than a random mammal and a random reptile.

In taxonomy (the system of biological classification), monkeys (Simiiformes) are an infraorder of the order of primates (Primates). Apes (Hominoidea) in turn are a superfamily within the Simiiformes. It's an "every thumb is a finger, but not every finger is a thumb" situation.