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A vowel is a sound generated by an open vocal tract, with vibration of the vocal cords and without friction. A consonant is every sound that is not a vowel. These two concepts are very simple and clear.

However, the whole concept of a semivowel seems to me a consequence of the false statement that the beginning sound and the final sound of a syllable (respectively, an onset and a coda) must be consonants. In several words of several languages, a diphthong (eg. "pay", "fly" and "yes" in English) or a triphthong (eg "Paraguay" and "why" in English) may begin/end a syllable. As far as I can hear, there isn't any difference in pronunciation of these vowels when are in the beginning or end of a syllable and when they are in the middle of it (nucleus). AFAIK, for instance, [flaɪ̯] and [flaj] are pronounced identically. Therefore, why are there different IPA symbols for semivowels (eg [w] and [j]) ?

Alan Evangelista
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There are a few different and mutually incompatible definitions of "consonant" and "vowel". One is, like you said, that vowels have no friction. But what about approximants like [l] or [ɹ]—or, for that matter, [j] or [w]? Those sounds have no friction and no complete closure, so wouldn't they be vowels?

Another definition is that vowels form syllable nuclei, while consonants don't. (More often this is called "syllabic" versus "non-syllabic" for clarity.) This notably requires defining what a "syllable" is; like phonemes, syllables aren't anything that can be quantitatively measured on a spectrogram, they're theoretical constructs that make phonological theories more elegant. And not all theories use them.

But many theories of phonology do use syllables, and in practice this syllabic/non-syllabic distinction turns out to be extremely useful for those theories. [i] and [j] in theory have exactly the same formants, it's just that the former is a syllable nucleus, and the latter isn't. Same with [u] and [w], and [y] and [ɥ], and [ɑ] and [ʕ].

As for why there are different symbols? Historical artifacts, really. The IPA vowel chart and consonant chart are completely separate and mutually exclusive, so when the same phonetic sound can be both a "consonant" and a "vowel", they had to make separate symbols for them. More recently, diacritics have caught on to switch one to the other, which is why we talk about English /n̩/ and Latin /e̯/ instead of making up entirely new symbols.

EDIT: It's worth noting that, in practice, semivowels do also tend to be a bit closer than actual vowels. This is usually considered a phonetic detail and ignored by phonologists.

Draconis
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    It might be worth noting that while "in theory" semivowels have the same formants as the corresponding vowels, in practice, in a few languages where the semivowel symbols tend to be used, there is, or there is thought to be, a narrower closure of the vowel tract in the semivowels compared to the corresponding vowels. Of course, though, you're not wrong in saying that "in theory" they are the same, since that's how the IPA defines them. – LjL Jul 01 '19 at 17:52
  • @LjL True! I'll add a note – Draconis Jul 01 '19 at 18:01
  • @LjL Could you give examples of languages in which this phonetic difference between vowel and semivowel exists? I do not hear it in English, German, Spanish, Italian, Portuguese and French. – Alan Evangelista Jul 01 '19 at 18:21
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    @AlanEvangelista To use one of my new favorite minimal pairs, compare English "eat" and "yeet". Your mouth opens slightly as you transition from the [j] to the [i] in the latter. – Draconis Jul 01 '19 at 20:49
  • @AlanEvangelista I think it happens in most languages that have these semivowels, but it's not something you typically "hear", it more just shows up in sound analyses. I'm sorry, I don't have sources to provide about that, it's just something I remember reading more than once. The kind of example Draconis gave is good to be able to "hear" it when next to a supposedly-identical-but-not segment. Note that in Spanish, intervocalic /j/ is often realized in a number of ways that are clearly distinguishable from [j], such as [ʝ], or even [ʃ] in Argentina. The Italian version is much closer to [i]. – LjL Jul 01 '19 at 21:09
  • @Draconis I do not know the word "yeet", therefore I am not sure if I know how to pronounce it, but my first guess is [i.it] or [i:t]. I can not perceive a change in mouth opening in any of them. – Alan Evangelista Jul 01 '19 at 21:34
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    @AlanEvangelista Ah, alas. Try "east" versus "yeast". For me at least they're quite distinct, with the [j] being closer than the [i]. "Wu" versus "ooh" is less distinct but also noticeable. – Draconis Jul 01 '19 at 21:44
  • @LjL the multiple renderings of /j/ in Spanish are out of the scope of this discussion IMHO. You could write /desaiuno/ instead of /desajuno/ as phonemic transcription and still have different allophones of /i/ (when they precede a vowel in the same syllable) in different dialects. btw I have a personal opinion that phonetic transcriptions are much better than phonemic ones in language learning because they reflect more explicitly the details of each dialect, but that's off-topic too. – Alan Evangelista Jul 01 '19 at 21:45
  • @AlanEvangelista if you only accept renderings of [j] that are not phonetically different from [i] due to those being irrelevant allophones, then I'm afraid, indeed, I cannot offer you languages, or dialects, where [j] does sound different from [i]. Sorry about that. In general, anyway, I do fully expect that especially in languages that can't typically have a [ji] segment, the difference between [j] and [i], even if it's there when you look at the formants, won't be "heard" by pretty much anyone who uses those languages. – LjL Jul 02 '19 at 00:38