Taiwanese, Cantonese, Mandarin and other Chinese dialects are kind of like sister languages. They are called “dialects” rather than separate languages because there exists a more-or-less uniform writing system that sort of unifies them. People spend a lot of time arguing about whether they’re really separate languages or dialects, but frankly I could care less.
The relationship between them is rather like the relationship between Spanish, French, Catalan and Italian. French would be in the position of Taiwanese: it split off from Latin earlier (I think…can’t remember exactly) and so it retains some earlier Latin-y stuff (if this is coming back to me correctly). Taiwanese is a more conservative dialect than Mandarin (note the greater number of tones, the final consonants allowed on syllables, stuff like that; these are features that were seen in Ancient and Archaic Chinese, and are retained in Cantonese and Taiwanese, but not Mandarin (only final consonants allowed in Mandarin are nasals n, ng).
A prof I once had commented that Tibetan decided to go with a super-wacky alphabet (they stack consonants three and four high) to produce words that sound alike but are written differently; Chinese, on the other hand, went with characters. Not sure which makes life easier (I studied Tibetan briefly in the States – hearing that monk spell his way down those consonant stacks was an experience I assure you).
The basic relationship between Taiwanese and Mandarin are as sisters. However, like all siblings, they’ve had different experiences, and they’ve been exposed to different things, so they’ve picked up some baggage from other places (in Taiwanese, most notably from Japanese for obvious historical reasons).
Taiwanese has not “adopted” Chinese characters. The factions for “how we should write Taiwanese” are still at war, with some preferring Church Romanization, some a scheme using tonal spelling, some others preferring this system that uses silent symbols to indicate some kind of meaning in a Romanized word (I couldn’t quite get my mind around THAT one, personally), and others preferring to use Chinese characters, although with some additions. Anyone who wants to see this can go to the “tai wan e tiam” store on the first alley above McDonald’s across from Taida…they sell tons of Taiwanese books, and their sign has the character for “e” (which is kind of like the possessive “de” in Mandarin) which looks kind of like a drunken “xia4” for “down” in Mandarin. Hard to describe.
It’s fun, however, to read Taiwanese written in characters because people who don’t know that Taiwanese can be written with characters can’t figure out what the heck it says in many cases. I was on a plane once next to a Chinese guy and I had a book by Robert Cheng, who does research on Taiwanese. It just about drove my seatmate crazy, and he finally had to ask what kind of book it was because his sidewise glances weren’t enough to figure out what language it was!