There are mainly two differences as far as I know. One is in wrting and the other is in speaking. First, Traditional Chinese Character is used in Taiwan while Simplified Chinese Character is used in the mainland. What's more, there are many differnt names of the same subject for the two areas. One example for bike is given by the first comment. In addition, there are also some differences in the pronunciation for the same words.

But they are all Chinese, so in most cases, they can understand each other easily. Because the language used in Taiwan is originated in the dialect of Fujian which is in the south of China.

PS: according to the history and the law of China, Taiwan is always a part of China. So it is more appropriate to change your title into 'How different is Mandarin used in Taiwan and the mainland?'.

I have lived in both China and Taiwan as a language learner; I've noticed some differences, but apart from the traditional versus simplified characters, these differences don't seem to be huge. It's mainly the occasional nouns and a few quirky differences in expressions. For instance Taiwanese people often say "bu hui" for "you're welcome" whereas in China they may more often say 'bu ke qi' or "mei shi." But anyway these differences in tendency seem to be almost always understood by people from both Taiwan and China. Sometimes native speakers will say to me, "They use that word in China, but here in Taiwan we say X" and vice-versa. 

As for Ella's comment regarding Taiwan-China relations, I know of several people from Taiwan who would point out that the "law" in China does not necessarily settle complex issues of national sovereignty and cultural identity. They would strongly object to such simplifications and logical non-sequiturs. 

Coming back to the actual topic though, I've seen this blog entry which seems to be a nice of summary: https://hcyip.wordpress.com/2013/11/02/beijing-life-mandarin-differences-between-the-mainland-and-taiwan/. ;

在台灣，bicycle最口語都是說 腳踏車

There's no big difference between the 2, just like British English and American English, French French and Quebec French. People speak same language but use different words in some contexts.We can tell the difference right away but to non-Chinese speakers, it's very hard to tell. I haven't had any problems communicating with taiwanese.

actually, in China has many different accent as local language with different regions..

I'm from east-north area of China, I completely don't understand about Shanghainese and contonese and other south local language.

haha,

Deo.