What's the difference between Mandarin Chinese and Taiwanese?

I don’t want to be ignorant and offend people by saying that they are the ‘same thing’ just how people say that hispanics and mexicans are the ‘same thing’ which they aren’t. So with the Mandarin Chinese and Taiwanese what are their differences and similarities? Is the language and the writing system the same or not?
PS:
If I spelled mexican wrong I’m sorry that tends to involuntarily happen when I type fast…

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