Why "planet of the apes" but "planet of the blind"

Solution 1:

One of the confusing issues here is the unusual noun phrase the blind. This noun phrase is unusual because, well, the noun phrase has a determiner and an adjective, but has no noun in it. It is an example of what grammars such as the Cambridge Grammar of the English Language call a fused modifier-head NP (noun phrase).

These types of noun phrase have an adjective functioning as Head of the phrase. They occur with a very small range of Determiners, primarily with the definite article the. They usually have a generic meaning, and thus when referring to countable things have a plural interpretation. Here are some examples:

• The rich
• The good
• The poor
• The infirm
• The Dutch

We always see plural verb agreement when these generic fused modifier-head NPs are Subject:

• The good are good, but the bad are better.

There are one or two instances where these NPs may be definite:

• The accused
• The deceased
  _{(These two Examples from CaGEL p. 418)}

In such instances the NPs can have singular or plural interpretations:

• The accused is required to present herself before the magistrate.
• The accused are required to present themselves before a magistrate.

Notice that in all of these examples, because the Heads of the phrases are adjectives and not nouns, there is no plural inflection present within the NP.

The Original Poster's Question

Both The Planet of the Apes and the Planet of the Blind contain semantically plural noun phrases as the Complements of the preposition of. It would certainly be possible to have a title:

• The Planet of the Ape

However, although very similar in meaning, this would have a slightly different connotation from Planet of the Apes. In this title the Ape would refer generically and abstractly to the species rather than referrentially to a bounded group of individuals. In this case the noun phrase the ape would take singular verb agreement:

• The ape is a sociable animal.

There is no way of doing this with the fused modifier-head NP, the blind:

• *The blind is eligible for extra assistance when filling in these forms. (ungrammatical)

Solution 2:

The answer to this question

Is that "s" necessary after the ape?

is no, the ess is not necessary. "The planet of the ape" is well-formed, and doesn't necessarily refer to a planet with only one ape. It might, instead, refer to a planet dominated or overrun by apes, for example (other examples are possible, including 'a planet that was at one time notable in some way for apes'). 'Ape' may be used as a mass noun (or, more properly, a non-count noun, see quote from Encyclopedia of Language and Linguistics below) denoting, generally, the quality or notion of some sort of 'apeness' attributed to the planet.

One passable explanation of how this works may be found at "Is the countable vs mass noun distinction common outside English?":

In English, there are a lot of nouns that are comfortable occurring as either mass or count, the difference ending up in whether we want to express a notion of the material, [ape], or that of some quantity of individuals comprised of that material, [apes].

As for this question,

Why is it planet of the apes but planet of the blind?

The plural noun, 'apes', may also be used. The signification of 'apes' differs from both 'ape' as a non-count (mass) noun and 'ape' as a singular noun. Notably, 'ape' in "the planet of the ape", in the case where the signification is not singular, is neither singular nor plural:

Whereas linguists often distinguish ‘count nouns’ and ‘non-count’ nouns, philosophers more commonly distinguish ‘count nouns’ and ‘mass nouns’ — a dichotomy often thought to have ontological significance. Count nouns are supposedly distinguished semantically from mass nouns by some criterion of ‘divided reference’ (‘criteria of distinctness,’ ‘individuation,’ etc.). But the supposed criterion is unsatisfactory, actually distinguishing singular count nouns from both plural count nouns and non-count nouns alike. By contrast, the count/non-count dichotomy — which is purely semantic, having no ontological significance — is not similarly flawed. Significantly, since count nouns are semantically either singular or plural, to be non-count is to be neither singular nor plural.

(From Encyclopedia of Language and Linguistics (Second Edition), "Mass Nouns, Count Nouns, and Non-count Nouns: Philosophical Aspects", H. Laycock, 2006, Pages 534–538. Emphasis mine.)

Solution 3:

Ape is a noun, (specifically a countable noun as mentioned in @CandiedOrange 's answer) and you add an "s" to turn it plural.

Blind is an adjective. When one says "planet of the blind" the noun is implicit and must be ascertained from the context. If the noun were included, then it would be plural. (I.e. "planet of the blind apes")