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 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 thus not the same phonemes.
Edit: The set of American English phonemes (from Kenyon and Knott) is available here.