what's a dialect?

In Taiwan, the native dialect is Taiwanese, and the official language is Mandarin Chinese. (Although, there is a push to make English an official language as well.)

Taiwanese and Mandarin are similar in syntax, but very different in pronunciation and intonation. Someone who understands Mandarin will not understand Taiwanese. For the most part, words can be translated directly, and are represented by the same characters in writing, but direct translations often sound awkward and sometimes are incorrect. For example, the word for "run" in Taiwanese corresponds to the word for "walk" in Mandarin.

The other day, I was talking to someone about the distinction between dialects and languages. My argument was that Romance languages such as French, Spanish, and Italian are just as similar (perhaps more so) than different dialects of Chinese such as Taiwanese, Cantonese, and Shanghainese. Her comment was that all Chinese dialects use the approximately the same written characters to represent the same words, which sounded reasonable to me.

But, why are Spanish and Italian written differently? It comes down to the usage of a phonetic alphabet. I bet that historically, words in all Romance languages sounded almost the same. As pronunciation began to diverge, spelling would also be forced to evolve. For example, Old English is significantly different from Modern English, even in writing, to the point where it is unrecognizable by native English speakers. In Chinese, characters represent whole words or parts of words, and they aren't constructed phonetically. Thus, different "dialects" can share the same characters. (For now, I'm ignoring the difference between the "Simplified" characters used in China and the "Traditional" characters used in Taiwan.) In fact, even Japanese kanji are just a subset of Simplified Chinese characters.

So, to go back to the initial question: What's a dialect? What's a language? Where do we draw the line?