The difference in spelling and pronunciation exists because "Taoism"/"Daoism" are formed from a Chinese word which can be transliterated either as "Tao", using the older Wade-Giles transliteration system, or "Dao", using the newer and more preferred pinyin scheme for the romanization of Chinese. "Taoism" is the older and more common spelling of the word, which appeared first in English in 1836 (according to the Oxford English Dictionary); "Daoism" is a newer variant which, like "Dao", follows the pinyin scheme.
Due to fundamental differences between Chinese and English phonology, neither /d/ nor /t/ can be considered adequate representations for the consonant at the beginning of the word "Dao"/"Tao". Either way to transcribe this consonant in English is merely an approximation; it so happens, however, that the correct pronunciation of the Chinese word is more likely to be perceived by a native English speaker to correspond to the spelling "Dao" than to the spelling "Tao". Thus many argue that the English pronunciation "Daoism" is closer to the Chinese pronunciation of the word dao than the English pronunciation of the word "Taoism" is. This example illustrates one of the several advantages that the pinyin transliteration scheme has over the older Wade-Giles scheme, from a scientific, linguistic point of view.
Some people think that existing words in English which come from Chinese words should be remodeled after the pinyin transliteration scheme, in light of its several important benefits over older transliteration schemes; other people think that the older forms should continue to be used because they are words in English, and not in Chinese anymore, while new borrowings should be written according to the better transliteration scheme.