The difference in spelling and pronunciation exists because "Taoism"/"Daoism" are formed from a Chinese words which can be transliterated either as "Tao", using the older Wade-Giles scheme, 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 follows the pinyin transliteration scheme for Chinese.
Many also prefer "Daoism" on the grounds of pronounciation. Whilst the Chinese sound represented by the pinyin d or Wade-Giles t is neither the English d nor the English t, the English d is a closer sound to it than the English t is. Thus many argue that the English pronounciation "Daoism" is closer to the Chinese pronounciation of the word dao than the English pronounciation of the word "Taoism" is.
Some people think that existing words in English which come from Chinese words should be remodeled after the pinyin transliteration scheme, which has 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.
See also: Taoism -- Wikipedia commentary/Use pinyin not Wade-Giles
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