The Republic of China (ROC) and Taiwan are practically synonymous today, since the ROC is currently the official name for the government of Taiwan. However, this was not always the case. As the name suggests, the ROC used to be the ruling government of China, based in Nanjing.
Here’s an example for the difference in history between Taiwan and the Republic of China:
During WWII, the Republic of China was one of the main powers of the Allied Forces, while Taiwan was under the Japanese Empire and many Taiwanese were conscripted to fight for the Axis Forces.