The original Planet of the Apes still stands because it did so many things right. First, it worked on two levels: both on the surface story and as an allegory of the times. In 1968 the country was going through a tremendous crisis: we had the assassinations of Martin Luther King and Robert Kennedy, the Democratic Party convention in Chicago was thrown into chaos because of the riots outside and the Vietnam War plank of the party platform inside, and the war itself was fiercely dividing the nation. The issue of civil rights was also not settled to say the least.
Into this maelstrom of a year entered Planet of the Apes. It can be seen an appreciated just for the story we see: a human who finds himself in a world where it is apes that are the dominant species and humans the servile animals. However, it is easy to see behind the story presented to us. 1968 also spoke, in a clever and subtle way, about the situations America was facing. The issue of civil rights is there in the film: the orangutans (who, by coincidence, are the lightest-skinned apes) are the ones who have all the power, while the gorillas (coincidentally, the darkest apes) are the soldiers and those who do the most menial work. Even some of the dialogue addressed race relations: at one point the female chimp Zira tells Taylor (Charlton Heston) that to apes, all humans look alike. How often and sadly has that mindset taken place among people of various races?
1968 also addressed the growing war in Vietnam: the idea of how the apes (specifically the gorillas who thirst for war) want to go to war against the humans (specifically in a jungle-like setting) could be transferred into our adventures into the jungles of a civil war between the Vietnamese. Though this idea would be explored further in Beneath Planet of the Apes the seeds for a commentary about war were already there. By placing Heston (who has always been identified with American strength in films) as the weakest character, unable to triumph over the animals, he appears to be unwittingly prophesying how our actions in Vietnam would leave our powerful nation weak and torn.
Finally, the film addresses the still-continuous battle of evolution vs. creation. The apes believed that The Lawgiver made Ape in his own image, so the proof of a pre-Simian society must be destroyed in order to preserve the peace of the Ape world. By his mere existence, Taylor strikes fear into the educated and religious elites...and thus he is in danger of being exterminated.
In short, 1968 was a film with ideas that didn't overwhelm the story we were shown. A film that works on those levels makes it one that can be appreciated by those who want to watch a good action story (one that even has moments of humor--intended and otherwise) and those who like having a story with intelligence and substance.