As mentioned elsewhere,「台」and「臺」are different characters; the former is not a simplified derivative of the latter. In proper nouns such as「臺灣」, the first character is just a phonetic transcription character, and doesn't contribute to the meaning.
In orthodox Mandarin usage,「台」as a phonetic character is pronounced tāi, not tái, which is why Taiwan is "correctly" written as「臺灣」. Using「台」as a substitute for「臺」is a later vulgar innovation; they were pronounced differently. Names such as
are all pronounced "correctly" with tāi and written correctly with「台」, not「臺」, although nowadays you may see references to 三台 "incorrectly" pronounced as tái and using「臺」.
PRC totally merged「台」and「臺」with「台」, so there is no further discussion to be made for Simplified Chinese.
「台」and「臺」had different origins.「台」originally represented a word now written as「以」; note that the top of「台」is the same component as the left of「以」.
「以」(Zhengzhang OC: /*lɯʔ/ > MC: /jɨX/, Fanqie: 羊己切) originally depicted a person「人」holding an object.
Much of the time,「人」was actually omitted. This version is now written as「㠯」.
Towards the latter half of the Zhou Dynasty, a distinguishing mark「口」was added to「㠯」to create a synonym「台」, which is a representation of「以」widely seen before Unification by Qin.
Qin Dynasty forms added「人」back, but in a way disconnected from the object「㠯」. Our current shape of「以」is inherited from this form.
「台」lost the function of「以」and became used for other purposes. Orthodox definitions (inherited from later sources such as Shuowen Jiezi) records two main uses, both phonetic borrowings sounding similar but unrelated in meaning to「以」:
- OC: /*lɯ/ > MC: /jɨ/, Fanqie: 與之切 > Pinyin: yí
- Original form of「怡」
- OC: /*l̥ʰɯː/ > MC: /tʰʌi/, Fanqie: 土來切 > Pinyin: tāi
The second pronunciation led on to its vulgar usage for tái, now unofficial in Taiwan and official in PRC.
「臺」originally represented a word that meant look-out tower/platform/terrace, and still retains this meaning today.
「臺」(OC: /*dɯː/ > MC: /dʌi/, Fanqie: 徒哀切 > Pinyin: tái) was originally comprised of semantic「宀」(building) and phonetic「之」(OC: /*tjɯ/ > MC: /t͡ɕɨ/, Fanqie: 止而切 > Pinyin: zhī).
The semantic component「宀」was later transformed into a combination of「高」and「室」, leading on to the modern form「臺」.
Following the Zhengzhang reconstructions, what is now Pinyin [y] (OC: /*l/) of「台」was connected to [t] (OC: /*d/) of「臺」through an aspirated voiceless alveolar lateral approximant /*l̥ʰ/; without tracing their evolution, Pinyin [yi] and [tai] do not really resemble each other.