I don't believe that people are nearly so afraid of being closely related to apes as they are afraid of having their long-held religious beliefs dispelled. The theory of evolution is just that - theory. It hasn't been proven beyond doubt. Religious apologists hold on to that for dear life because they know that if Darwinism were proved without doubt, then all that time clinging on to the childishness that is spewed out in the Bible was wasted. If this sounds unfairly harsh towards those who want to believe the Bible verbatim, it really isn't any harsher than those who mockingly tell us who trust in Darwin, "I ain't descended from no apes."
The truth of the matter is, if you look at the science objectively, the evolution theory makes much more sense than anything Bible-based. Why can't it be proven beyond doubt? Even though the progression from one-celled organisms to smaller complex life forms to fish to land animals and up the chain to what's now considered mankind is a logical one, there can always be the question of where the first amoebae (or other one-celled life forms) came from. Those who trust the Bible say that someone or something has to have started the whole thing and that someone/thing is God. Of course, the atheist's reply is then, well who started God? The religious would then reply, God just always was. My response to that is, if you can believe that God just was, why can't you believe that the universe just was? This isn't even necessarily anti-religious. Typically, this is the Buddhist/Hindu type of belief. God and the universe are one and the same. They've just always existed. But, firm believers in Adam and Noah and Moses don't want to hear it.
Then there is the theory of "Intelligent Design." While this theory tries to appease both sides of the aisle, the real truth is, intelligent design is nothing more than Bible thumping under a different name. Essentially, believers in this theory state that, yes, some things may have developed, changed and evolved, but the whole thing was sparked by God in the beginning. While this may sound like a compromise at first, the fact is intelligent design apologists will never discredit the silliness of the Noah's Ark story and will never even pay mind to the evolutionary theory. Intelligent Design is religion trying to pass off as science. And believers in Intelligent Design still want nothing to do with being descended from apes.
So, the question remains, "Why do people find it so unacceptable to be closely related to the apes?" The answer is very simple, really. If one buys into the theory of evolution, it means one has wasted a lifetime in other childish beliefs.