Xjyuk

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]) ?

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.