An affix is a morpheme added to some structure within a word, for example -ing is a suffix (post-root affix) and un- is a prefix (pre-root affix). An affix is an exponent of some form of inflection or derivation added to some word category. A clitic is more like an independent syntactic unit (i.e. a word on its own), except that it is always "dependent" in some sense on some other word. The literature of the past half century has not established that "clitic" is a coherent category, and many people hold that clitics are really something else.
Certain kinds of clitics are so analyzed because they have to go in second position in the sentence, for example the Lushootseed pronoun clitics čəd '1s', čəxʷ '2s' which come after the first word in the sentence. These "words", if they are words, can't be used alone (there are instead independent words ʔəca, dəgʷi) – in this sense, clitics are like affixes. But an affix typically goes on a certain kind of word, such as a noun, whereas clitics typically go in a syntactically-defined place and don't care what kind of word they go on.
The distribution of possessive -s in English is sort of a problem for many theories. It is glued on to the end of some word, but that word can be anything, because it goes at the end of the phrase marked "possessor". It appears on a noun in "The child's hand", a verb in "The child who ran's hand", a direct object pronoun in "The child who saw me's hand", and so on. And yet it is subject to a morphological restriction that it can't be added to a word with the s-plural ("The dog's paws" and "The dogs' paws" referring to 1 dog vs. plural dogs are pronounced the same). One theory of -s is that it is "a clitic", and another theory is that it is a "phrasal inflection", i.e. just another kind of affix (one with a peculiar distribution).
The question suggests that "plural possessive" is a formally distinct albeit homophonous suffix, just as the verbal 3rd singular present -s is formally distinct from the nominal plural -s, which you might argue for, but is not the only solution. Leaving out the possessives, the verb ending and the noun ending are both instances of affixation, and the difference between the affixes is what properties trigger insertion of one vs. the other (likewise, what properties trigger insertion of the suffix -d in verbs, what triggers the suffix -ing, and so on). When the form and function are clearly distinct, we have no problem saying that there are two suffixes.
When two or more functions seem to have the same form, it becomes more problematic to say "clearly there are two affixes". Most people accept the existence of a verbal inflectional suffix -s and a nominal inflectional suffix -s because their functional properties are pretty much disjoint. Analysis in terms of functional properties is very dependent on the theoretical framework that you are using. You could assume some set of features for words like noun, verb, singular, plural, possessive, 3rd person and so on. One framework might take it as obvious that "plural possessive" is an inflectional category of English, and another might conclude that it isn't, and that there is another explanation for the lack of double-suffixation. Without a more specific theoretical framework for syntax and morphology, it's impossible to say how one should handle the plural-possessive problem.