What is the difference between the vowel in the word "cup" and the first vowel in the first syllable of the word "about" in terms of phonetics? Why do they have to be shown in different symbols? Do they sound the same?If we are referring to RP and the kind of pronunciation usually taught to EFL learners who take BrE as their model, then the vowel of cup (aka the STRUT vowel) is considerable more open than that in the first syllable of about (aka “schwa”). That is, it is more similar to Turkish [ɑ].
They are represented by different phonetic symbols, [ʌ] and [ə] respectively, because they sound different and, more importantly, because there are a few pairs of words or phrases in which the difference is important for the meaning. For example, unorthodoxy [ʌnˈɔːθədɒksi] sounds different from an orthodoxy [ənˈɔːθədɒksi]. A hubbub is always a [ˈhʌbʌb], never a *[ˈhʌbəb]. A large untidy room [ə ˈlɑːdʒ ʌnˈtaɪdi ˈruːm] is different from a large and tidy room [ə ˈlɑːdʒ ən ˈtaɪdi ˈruːm]
There are only a few such cases, though, because the STRUT vowel belongs to the strong vowel system and is often stressed, while schwa belongs to the weak vowel system and is never stressed.
In other varieties of English the picture is different. There are many English and Welsh people who pronounce the STRUT vowel pretty much the same as schwa. So do many, perhaps most, Americans. For them, we could reasonably write both vowels as [ə]. So it would be reasonable for EFL learners not to worry too much about the contrast. Millions of NSs manage without it.
Your textbooks will tell you to pronounce above as [əˈbʌv] and another as [əˈnʌðə]. But millions of people in Birmingham, Swansea or Miami say [əˈbəv] and [əˈnəðə(r)]. You can too, if that’s easier.
But not if you want to speak RP.