Claimed by China, Taiwan has repeatedly attempted to join the UN but has been vetoed each time. Do they deserve recognition as a country or are they a mere province?
Years later, Quebec whines about loss of culture, lack of power in the Canadian government, they dislike the power of the federal government, because they see it as English.
The Peoples Republic of China ie PRC ie China is a communist/socialist nation The Republic of China ie ROC ie Taiwan is a democratic nation.
Therefore Taiwan really doesn't want to be absorbed back into China. With the U.N.'s adoption of a one China rule and the PRC being officially recognized by the U.N. Taiwan got informally screwed.
Oh and I forgot one more important part of this. Both sides agree that there is only one China so the PRC and the ROC will not let you recognize both. And since the ROC does not have the economic power of the PRC, they are losing out heavily. But I hope this helps answer why the U.N. doesn't recognize Taiwan.