My analogy is not biological. Latin and Italian are separate languages, yet one evolved from the other. If the difference is not big enough for you, then use Italian and Proto-Italic, or Proto-Indo-European. The point is to illustrate evolutionary continua.
Additionally, humans are still apes, even paraphyletically speaking. And when you say "many DNA similarities," remember that, with chimps for example, we're talking a 95%< identical DNA kind of similarity. Also, apes aren't a species, what you are referring to as apes comprise more than one genus. Indeed, chimps are more closely related to humans than to gorillas, gorillas are more closely related to humans than to orangutans, and orangutans are more closely related to humans than to gibbons, etc... As for evolutionary continua, such continua actually exist in nature, and macroevolution has in fact been observed on many occasions. But that's only on the species level; evolution doesn't give you big jumps such as between humans and other comparable modern anthropoids within a matter of mere centuries.