Besides monkeys there are other types of primates. A suborder of primates, Haplorrhines are all monkeys and these weird bug-eyed things called tarsiers. But there is another suborder called Strepsirrhini. These are all lemurs, lorises, and galagos. Extinct taxa include Omomyoids and Adapoids, which are probably early evolutionary forms of Haplorrhines and Strepsirrhines respectively.

Monkeys are everything in Haplorrhini, sans Tarsiers. This means new world (but less derived) monkeys like capuchins, old world monkeys (but more derived) like baboons and macaques, and apes (which I include humans in).

So, it isn't incorrect to call chimps et al. apes, but it is incorrect to say they're not monkeys. Clades are umbrella terms, they start at a point and encompass everything beyond that point.

I am most interested in strepsirrhines but I keep up with monkeys-formal, so I say I work with primates.

Here: Primates includes Strepsirrhini and Haplorrhini. Streps can be subdivided into lorisiformes and lemuriformes. Haplorrhini can be subdivided into tarsiers and monkeys-formal; monkeys-formal into old world monkeys-formal and new world monkeys; new world monkeys into a bunch of genera and old world monkeys-formal into old world monkeys (cercopithecoids) and hominoids; hominoids into lesser and greater apes; greater apes into pongines and hominines. Each level encompasses everything below it. Both pongines (orangs) and hominines (gorillas, pan, homo) are a part of the great apes clade. Both the great ape clade and lesser apes are a part of the hominoid clade. Hominoids and cercopithecoids into the OWM clade. OWM and NWMs into the monkey clade. Monkeys and tarsiers into the haplorrhine clade. Haplorrhines and Streps into the primate clade.

See how nicely it works out when things are cut clear? If we are not aware of the issue of paraphyletic clades then nomenclature becomes an unnecessarily confusing issue. Suddenly we have a clade of monkeys which ignores its evolutionary descendants. But we're working within the paradigm of cladistics (or phylogenetic taxonomy) in which evolutionary descent is all we're worried about! So what good is this clade if it just confuses matters? So I'm pushing for a correct nomenclature in which we shed paraphyletic clades (unless they achieve a specific end in which they should be used very consciously).