It's beyond accent. There are at least 7 variants of Chinese that are as different as French and Spanish. Chinese variants share a common written language, but the spoken languages are unintelligible to people who speak another variant. Most on the mainland speak Mandarin. Most in Taiwan speak Cantonese.
The first part is correct. In fact Mandarin has 4 tones as part of the language (flat, rising, falling, and a fall/rise) so a flat tone “ma” means mother and a falling and rising tone on “ma” means horse. Catonese has 5 tones. There was a US general in China during WWII that helped communicate between the northern and southern army because he knew both spoken languages.
The Taiwan part is Incorrect. Cantonese is mainly spoken in south mainland and Hong Kong.
Taiwan has 2 main languages - a local island Taiwan dialect and Mandarin from all of the ROC escapees after the PRC took over the mainland.
My Chinese prof at USNA was an escapee to Taiwan... he worked with the OSS during WWII and at one point was a TV reporter in Taiwan + other stuff after the war until he came stateside.
One other note is that the mainland implemented a simplified form of their characters (hanzi) for the written language. They say it was to speed learning for the masses - which may have been partially true - but I think it was also to prevent the masses from learning or reading from past writings. Taiwan, Hong Kong, Singapore, and many other Chinese enclaves around the world continued to teach the classical characters. So now the written language has begun to separate as well.