Short answer: identical. Taoism and Daoism refer to the same tradition, the same founder, the same book. The word is 道, and the difference is entirely a matter of how you spell a Chinese sound using the Latin alphabet.
And the pronunciation is dow, rhyming with "now", in both spellings. "Tao" was never meant to be said with an English T.
Wade-Giles and Pinyin
Wade-Giles was developed in the nineteenth century by Thomas Wade and refined by Herbert Giles, and it dominated English-language writing about China for about a hundred years. It uses an apostrophe to mark aspiration: t' is the breathy T in "top", while plain t is the unaspirated sound that English ears hear as a D. So Wade-Giles "tao" always meant the D-ish sound — the apostrophe that would have made it a real T is absent, and that absence is doing all the work.
Then the apostrophes got dropped in newspapers, on book covers and in conversation, because typesetters and readers found them fussy. A century of English speakers duly said "Taoism" with a hard T, on the reasonable assumption that the spelling meant what it looked like.
Pinyin was developed in mainland China in the 1950s and adopted internationally around 1979. It spells the sound as it is: dao. It is now the standard system in scholarship, journalism and Chinese teaching.
The full swap table
Wade-Giles on the left, Pinyin on the right. Same word each time.
Tao / Dao · Taoism / Daoism · Tao Te Ching / Dao De Jing · Lao Tzu / Laozi · Chuang Tzu / Zhuangzi · Chi / Qi · I Ching / Yijing · Confucius, oddly, is neither — it is a Latinisation of Kong Fuzi produced by Jesuit missionaries in the sixteenth century.
So which should you use?
Use Pinyin when precision matters — scholarship, anything involving actual Chinese, anything where a reader might try to pronounce it. Dao, Laozi, Zhuangzi, qi.
Taoism is still fine in general English, and we use it, for the unglamorous reason that it is what English speakers search for and what almost every English edition on a shelf says on its spine. The Tao Te Ching is a book title that has been printed that way ten thousand times; changing it on a book cover mostly loses the reader. Language is a used object, not a corrected one.
The one thing worth fixing regardless of spelling is the pronunciation. It is dow.
The other spellings you will see
Daoism, Taoism, Tao-ism, and then the genuine misspellings that turn up constantly in search: daosim, doaism, taoizm, dioism, douist. All of them mean this. If you arrived here having typed one of them, you were not wrong about anything that matters.
What to read next
Now that the spelling is out of the way: what the Tao actually is, whether Taoism is a religion (the answer is genuinely both), who Laozi was and whether he existed, and the symbols. For the book itself, which translation to buy matters more than the spelling on the cover, and 365 Days of Tao (ours, flagged) takes it one verse a day across a year.
Common questions
Is it Daoism or Taoism?
Both, for the same tradition. Taoism is the older Wade-Giles romanisation, Daoism the modern Pinyin one. Pinyin is standard in scholarship; Taoism remains common in general English and on book covers. Nothing about the tradition differs between them.
How do you pronounce Tao?
Dow, rhyming with "now". Not "tay-oh" and not with an English T. In Wade-Giles, an unaspirated t was written without an apostrophe and represents the sound English speakers hear as a D — the apostrophe would have made it a true T, and there isn't one.
What is the difference between Tao Te Ching and Dao De Jing?
Nothing but spelling. Same three characters, same text: Tao Te Ching in Wade-Giles, Dao De Jing in Pinyin. Roughly, the way of virtue's classic.
Why are there two spellings of Chinese words in English?
Because two romanisation systems were in wide use. Wade-Giles, from the nineteenth century, dominated English writing until Pinyin — developed in China in the 1950s — was adopted internationally around 1979. Books printed before then use the old system, and many titles kept it.
Is Laozi the same as Lao Tzu?
Yes. Laozi is Pinyin, Lao Tzu is Wade-Giles, and both are a title rather than a name: roughly "old master". Whether the person existed is a separate and much more interesting question.
What is qi, and is it the same as chi?
The same word. Qi is Pinyin, chi is Wade-Giles. Not to be confused with the chi in tai chi, which is a different character entirely — that one is ji in Pinyin, as in taiji.