Phonemics, or Phonology, is the study of the distribution of sound systems in human languages. A Phoneme is a particular set of sounds produced in a particular language and distinguishable by native speakers of that language from other (sets of) sounds in that language. That's what "distinctive" means -- the English phonemes /n/ and /ŋ/ can be told apart by native speakers of English, because we use these sounds to distinguish different words -- sin ~ sing, ton ~ tongue, run ~ rung, etc. This would be impossible if these phonemes weren't distinctive in English.
Phonetics, on the other hand, is simply the physiological and acoustic study of speech sounds, covering all sounds used in all languages, and relying only on the physical and physiological characteristics of the sounds, without regard to their systemic patterns in various languages.
Phonemes, the unit of (this variety of) phonemics, encased in /slashes/, are always specific to a language. Since phonetics is a natural science, phones, the unit of phonetics, encased in [square brackets], are universal, and are not specific to any language.
Thus, we say that there is such a thing as "the phone [p]", because phones are defined universally, but that there is no such thing as "the phoneme /p/", because phonemes are relative to languages. Thus "the French phoneme /p/" and "the English phoneme /p/" both exist and are meaningful, and the phone [p] is represented in both of them; but they are not the same sets of sounds and they don't have the same distribution, and thus are not the same phonemes.
Edit: The set of American English phonemes (from Kenyon and Knott) is available here.
It turns out that writer and rider are not “indistinguishable” in much of the United States. The difference is that although both rider and writer have an alveolar flap in their middles, writer is [ˈɹʌɪɾɚ] with a raised and somewhat shortened diphthong, whereas rider is simply [ˈɹaɪɾɚ].
The original /t/ of write was enough to trigger so-called Canadian raising in the diphthong, since write has it but ride does not. This distinction is preserved in the longer versions ending in ‹r›.
Understand that although the phenomenon is called Canadian, it is by no means limited to that country, but extends to much of the United States as well.
We don’t think of the raising as being phonemic, but it is enough to disambiguate what would otherwise be homophones. In the referenced Wikipedia article, they also point out that this is what allows us to distinguish high school (the one after junior high) from a high school (one that is high).
Best Answer
They're told apart by the same way that /f/ and /v/, or /s/ and /z/, are. You use your vocal cords for /v/, /z/, and /ʒ/, but not for /f/, /s/, /ʃ/. Aside from that, they're identical.