Talking about pronunciation and vocabulary, there aren't many differences between Taiwan and Mainland. You can compare them with British and American English. A fluent Mandarin speaker has no difficulty communicating with people from both regions. 

While the writing system is greatly distinct. Mainland uses a simplified version of Chinese characters originated from Taiwan's traditional ones. Some of the characters are written the same while some are not, see this phrase which means "a melancholy turtle" written in both version of characters: "忧郁的乌龟(Simp.)", "憂鬱的烏龜(Trad.)". You may realize that the third character 的 are written the same and the fourth 乌/烏 are somehow similar. A native speaker can easily understand a different version of Chinese characters by the context but it could be hard for learners who already do not understand parts of the passage. Since the simplified version is widely used in Mainland, Singapore, Malaysia and many other place where there are Chinese people while traditional version is only used in Hong Kong, Macau and Taiwan, maybe you should study the simplified version if you want to communicate with more people from different parts of the world. However, if you like Taiwan movies and musics very much which makes you need to read things written in traditional Chinese, or you plan to know more about Chinese culture and study old Chinese(which is often written in traditional characters), maybe traditional Chinese is a better choise.