There are hardly any differences in grammar between them. The only difference I've noticed is that Taiwanese often use 有 before a verb even in interrogative sentences while mainland Chinese rarely speak that way. e.g. 你有買那本書嗎? 你买了那本书吗? The main difference between the two languages is the words and expressions coined in the last decades, many of which are translation from English. e.g. 软件 軟體 Software 操作系统 作業系統 operating system

Simplified symbols are used in Chinese while Traditional symbols are used either in Cantonese or in Taiwanese and there's almost no difference between Chinese and Taiwanese but the difference between Chinese and Cantonese is really huge

If you mean simplified and traditional Chinese characters, the difference is between their shape and strokes. If you mean modern Chinese and traditional Chinese, the difference is huge in grammar, vocabulary, pronunciation and a lot of aspects. It's kind of like a modern people reading Shakespear's works, you know that's English but it's not the English you are familiar with.

They are almost the same in both grammar and pronunciation, only very few differences exist. However, the expression habits of a same thing can sometimes be the major difference. For example, the word 'internet' is 互联网(互聯網 if in TChinese) in SChinese and 網際網路(网际网路 if in SChinese) in TChinese. the word 'panda' is 熊猫(熊貓 if in TChinese) in SChinese and 貓熊(猫熊 if in SChinese) in TChinese. It's very interesting because they are just inverted! and the word 'taxi' is 出租车(出租車 if in TChinese) in SChinese and 計程車(计程车 if in SChinese) in TChinese. ※SChinese stands for Simplified Chinese and TChinese for Traditional Chinese And there are so many such examples. However, if you use a specific S/TChinese noun to a T/SChinese speaker, he can still understand what you mean, only sounds a little weird.