The two are fundamentally different. It doesn’t matter whether anyone recognises Taiwan’s independence. Taiwan is effectively independent. You don’t need to apply to Beijing to get a visa to visit Taiwan. You don’t even need to ask Beijing for permission to conduct military exercises in Taiwan.
The obvious but often ignored fact is that Taiwan under the Republic of China has been an independent country since 1912. That’s longer than the history of the People’s Republic of China, PRC which has only been around from 1949. That is why the CCP was not recognised as the legitimate government of mainland China until Kissinger’s secret visit in 1971. Since the UN cannot recognise two Chinas, they decided to change their minds and recognise the PRC This piece of history is absolutely significant and cannot be ignored when we hear the ludicrous statement that the PRC does not recognise the “independence” of the ROC. If anything, it should be the other way around; that the ROC government does not recognise the legitimacy of the insurgents who had taken over the mainland.