Etymonline has this to say:
-ics
in the names of sciences or disciplines (acoustics, aerobics, economics, etc.) it represents a 16c. revival of the classical custom of using the neuter plural of adjectives with -ikos (see -ic) to mean "matters relevant to" and also as the titles of treatises about them. Subject matters that acquired their names in English before c.1500, however, tend to remain in singular (e.g. arithmetic, logic).
So yes, at some point in history, there were such things as physic (meaning "natural science"), mathematic (meaning "mathematical science"), etc. that were later turned into plural forms but kept being treated as singular.
Edit: having looked in a few more places, it appears that in contemporary English, it still makes some sense to have both the suffix -ic and its plural form -ics. According to the Collins English Dictionary, the former has kind of specialized in forming adjectives, while the latter is happily forming nouns:
-ic
suffix forming adjectives
- of, relating to, or resembling: allergic, Germanic, periodic. See also -ical.
[...]
[from Latin -icus or Greek -ikos; -ic also occurs in nouns that represent a substantive use of adjectives (magic) and in nouns borrowed directly from Latin or Greek (critic, music)]
[...]
-ics
suffix forming nouns (functioning as singular)
- indicating a science, art, or matters relating to a particular subject: aeronautics, politics
- indicating certain activities or practices: acrobatics
[plural of -ic, representing Latin -ica, from Greek -ika, as in mathēmatika mathematics]
The key here is that they are not just two unrelated suffixes. Much rather, one is etymologically a plural form of the other. As the American Heritage Dictionary succinctly puts it, -ics is "-ic + -s".
There is good discussion of the semantic differences in other answers, but the most important practical difference is that ignorant is a very insulting word that you should be careful about using, whereas naïve is not such.
In general naïve makes me picture a hopeful child who has unrealistic dreams and has not thought about the real world enough, whereas ignorant makes me picture a dumb, racist old man who won’t change his worldview in the face of overwhelming evidence.
So, for instance:
“I think you are being naïve, because...” is appropriate way to disagree with someone's theory at, say, a business meeting. It’s still a strong thing to say, and possibly condescending or belittling of your colleague’s theory.
On the other hand, “I think you are being ignorant, because...” is quite rude and aggressive. It's not practically very different from saying stupid even though the semantics differ.
Best Answer
Elizabeth Phrase in her book "English without tears" explains the difference between a little and little, saying they could even mean opposite things. She gives the following example:
She explains that this means that the person actually had an improvement as a result of the medicine he took, in contrast to "the medicine he took did him little good" which would mean the medicine actually worsened his condition.