I'm not sure it's true that all apes are monkeys, but I'm finding this more confusing than I thought it would be. I read the introductions to the Wikipedia articles on monkeys, apes and Old World monkeys, and it seems that New World monkeys are in one group, while apes and Old World monkeys are in another group, the Catarrhini. Apes are in the superfamiily Hominoidea, while Old World monkeys are in the superfamily Cercopithecoidea. I don't think Old World monkeys contains apes.
Apes are not monkeys. If you're speaking English.
That's because neither 'ape' nor 'monkey' are formally defined phylogenetic terms. They're common, everyday words which refer to groups of organisms we've lumped together for one reason or another
Hominoidea, Cattarhini, Platyrhinni and Simiiformes are formally defined phylogenetic terms. The realisation that apes are more closely related to babboons than to howler monkeys means that Hominoidea must be considered part of Cattarhini. It makes not a jot of difference to the meanings of ape or monkey.
All this confusion would be avoided if we weren't speaking English, since most European languages don't have two seperate words. In Dutch, a monkey is an aap, as is in an ape (of course, it's still paraphyletic used in the traditional sense, since humans are not aapen).
In Czech, there are different words for black rats and brown rats (krysa and potkan, respectively). Most rats are called krysa. But it makes no sense to therefore say that a potkan is a krysa since potkani are nested within the krysa clade.
Some normal terms are pretty much meaningless phylogenetically. 'Toad' refers to a wide range of anurans spread across the tree, and what's called a toad by some may be a frog to others. And of course, this distinction does not exist everywhere - there isn't a seperate word for frog and toad in every language.
This is why they invented the formal neo-Latin terminology in the first place.