There are many different kinds of Daoism and many different kinds of Buddhism. All with different views on important issues. Generally speaking Chinese Daoism and Indian Buddhism are different in their worldview, aims, and methods. However, Chinese Buddhism is massively influenced by Taoism and vice versa, over the entire history of Buddhism in China, so they can sometimes seem related.
The most salient different might be with respect to nature. Daoists aim to be in harmony with nature, through balancing out opposites, and through this to attain physical immortality. Indian Buddhists saw nature as the problem and escaping from rebirth in the world as the answer to this problem. On the other Buddhist these days are quite interested in environmental issues as an extension of Green politics and notions of compassion and are thus also quite interested in nature these days.
Daoists believe in a soul that survives death; Indian Buddhists believed that nothing like a soul can exist. Though arguably Buddhists capitulated to a soul when they invented "Buddha nature".
The methods of Daoism aim to bring about balance of the elements. The methods of Buddhism aim to help us see that our views about the elements are wrong and through seeing them correctly we can be free from being composed of them.
And so on.
However the massive influence of Daoism on Chinese Buddhism means that an understanding of Daoism is very relevant for understanding East Asian forms of Buddhism. The influence began at least with the earliest translations of Buddhist texts into Chinese. For example the character 道 was used to translate dharma and marga (way); and the character 神 is a term from Daoism that generally means “spirit”, “soul”, (sometimes translated as ‘divinity’). It is also used to translate Sanskrit ṛddhi “supernatural power”, anubhava “power, majesty”, or deva.
Later Daoism was profoundly influential on Zen Buddhism, and Zen has been one of the most influential forms of Buddhism in America (less so in Europe). According to one source Daoim introduced "elements like the concept of naturalness, distrust of scripture and text, and emphasis on embracing 'this life' and living in the 'every-moment'." Dumoulin, Heisig, & Knitter. Zen Buddhism: A History (2005).
Re spelling: "Daoism" uses the modern Pinyin system of transliteration for the Chinese character 道 i.e. dào; whereas the older and largely deprecated Wade-Giles system transcribes it tao. Apparently there is much discussion over which is "correct". I prefer Pinyin, but one can argue that Taoism is an existing Anglicisation that ought not to be affected by fashions in transcription.
The question has some crossover with others questions: