The "Republic of China" and Taiwan are not one and the same - this is something that both advocates of reunification and separatists agree on, but something that western supporters and detractors of China can't seem to understand.

Pro-PRC supporters of reunification view the PRC as the successor state of the ROC, and see Taiwan simply as a province that hasn't been reunified. This phenomenon is very common in Chinese history when one dynasty replaces another.

Separatists view the "ROC" government on Taiwan as a foreign regime imposed on the Taiwanese people. Hardliners seek to overthrow the "ROC" regime and replace it with a "Republic of Taiwan." Moderates are content with entering and desinicizing the "ROC."

Old-school pro-"ROC" types, which don't really exist anymore" view the PRC as an illegitimate power that occupies the Chinese mainland, and the the "ROC" is the sole legitimate government of all of China.

The PRC recognizes the reality that TW isn't governed by Beijing, but because the PRC considers itself the successor state of the ROC and that TW is a part of China (even according to the "ROC" constitution), it cannot recognize the regime on Taiwan as that of a sovereign country

It therefore refers to the "ROC" government as the Authorities of the Taiwan Area (台灣地區當局), and refers to territories under Taipei's control (Taiwan, Penghu, Jinmen, and Mazu) as the Taiwan Area (台灣地區).

To those who think Beijing denies the existence of Taiwan, what the hell do you think they call the island?

Beijing recognizing the fact that Taiwan is governed by remnants of the old Chinese government doesn't mean it recognizes Taiwan as a country - it just views the situation as an unresolved part of the Civil War - a domestic issue to be handled by the Chinese people on both sides

This is similar to how neither Pyongyang or Seoul recognize the existence of "two Koreas." To Pyongyang, there is a south Korea, but there isn't a "ROK." To Seoul, there is a north Korea, but no "DPRK."