Tao :

or Taoism (pronounced "Dow") is based on the yin-yang, which means opposites. It’s a Chinese religion based on a psychology and philosophy approach which dates back to 604 BC. The Yin (on the right) expresses everything that is light and good in the universe. In addition, the yang (on the left) expresses everything in the universe that in evil and dark in the dark in the world. According to the the Ying-Yang theory, you can’t have one without the the other. For instance, everything that’s born will eventually die, if you have day then you must have night, and if you have pain you must have joy. This religion focuses more on expression and experience rather than core beliefs.

The main focus on a path and this will carry you through life. It’s encouraged you focus on the Thi Chi, which is a form of exercise with slow, meditating movements. Taoism is a religion where its followers are taught to not accept the world’s pleasures and not strive for the best. Instead, it teaches its followers to be content with their lives. Taoists are believers that by nature everyone is a good person. In order to practice this belief, they treat everyone with kindness and love. They believe that every individual is born into purity and capable of doing good deeds.

Where as the afterlife is concerned, Taoists believed it shouldn’t be feared. They consider it part of the yang, just like life is part of the yin. Furthermore, Taoists consider the meaning of life to be constant discipline through diet, health and exercise. While this religion is centralized in China, it has followers worldwide. Across the world, the have over 20 million followers in Taoism.

Buddhism :

, which originated in India around 424 BC, has become a major world religion. Siddartha Gautama, an Indian prince, left his wife and son for six years to live an ascetic life. He starts to meditate and after he sees the light of truth, he turns in the Buddha. It’s during this time that he discovered the four noble truths: 1) Life is suffering; 2) Suffering is due to attachment; 3) Attachment can be overcome; and 4) there is a path for accomplishment. He also discovered the eight-fold path: It teaches followers right understanding, right thought, right speech, right action, right livelihood, right effort, right mindfulness and right concentration. All of these are taught through the art of meditation and exercise, similar to Taoism.

One key belief to Buddhism is the belief in karma. It simply states that your actions in this life prepare you for the next life. If you’re good and righteous, then in the next life you’ll be rewarded. However, if you’re evil and condemning, then you will be punished. Another belief they hold is that anything negative can be worked through with meditation, in the sense that meditation allows their followers to feel more positive and relaxed.

Whereas Taoists don’t fear death, Buddhists embrace death. Once you die then you are born again into a new life. Your actions in your new life will depend on how you behaved in the past life. These cycles will be persistent in continuation until the Buddha detaches their self completely from the world around them. The Buddha did not continue onto to another life because he became "enlightened," and this is the ultimate goal for all Buddhists. This is the fourth largest religion in the world.

Since we all do interpret the same facts differently, it will be up to you to decide if they are mutually inclusive or not.

Many people believed in both, and the mythologies and beliefs were freely mixed. For a very famous example, I'd point you to Journey to the West, basically a long Buddhist parable, which features a bunch of Buddhas and Bhodisattvas chilling out in Daoist heaven listening to Laozi give a speech, which was a fairly commonly described scene in old-timey Chinese literature. There are thousands of stories of various Xian hanging out with Buddist saints and joining forces and so on. I recently went to a temple dedicated to the very Daoist Shou that had dozens of Buddhist prayer wheels, where Buddhist ceremonies were performed when the temple was still active.

There were some points in Chinese history where Confucians/Daoists/Followers of Chinese mythology (those three being almost 100% mutually inclusive) rejected or occasionally suppressed Buddhism on the grounds of being foreign (for example, after the Anshi rebellion, started by a man of foreign descent and which inspired numerous foreign invasions hoping to attack China when it was weak, causing China to become extremely xenophobic), but those were for political reasons, not doctrinal reasons.

QUOTE: 

subhash.sah:

Siddhartha Gautama, not an Indian prince. Nothing gets on nerves of Nepalese more than when people don't get the facts right and say things like " Buddha, India" and "Mount Everest, China".

Siddhartha Gautama is born in Lumbini, near the Nepalese-Indian border.  But just as the year of his birth is to many unknown, some places within India, and some in Nepal claim to be its birthplace.  Actually, it is known that his mother gave birth to him while traveling from her family castle to her husband.s home.  So, your claim is as good as mine.

And according to Wilkepedia,  the most traditional biography, the Buddha's father was King Śuddhodana, the leader of Shakya clan, whose capital was Kapilavastu, and who were later annexed by the growing Kingdom of Kosala during the Buddha's lifetime; Gautama was the family name. His mother, Queen Maha Maya (Māyādevī) and Suddhodana's wife, was a Koliyan princess. Legend has it that, on the night Siddhartha was conceived, Queen Maya dreamt that a white elephant with six white tusks entered her right side, and ten months later Siddhartha was born. As was the Shakya tradition, when his mother Queen Maya became pregnant, she left Kapilvastu for her father's kingdom to give birth. However, her son is said to have been born on the way, at Lumbini, in a garden beneath a sal tree.

The day of the Buddha's birth is widely celebrated in Theravada countries as Vesak. Various sources hold that the Buddha's mother died at his birth, a few days or seven days later. The infant was given the name Siddhartha (Pāli: Siddhattha), meaning "he who achieves his aim". During the birth celebrations, the hermit seer Asita journeyed from his mountain abode and announced that the child would either become a great king (chakravartin) or a great holy man. By traditional account, this occurred after Siddhartha placed his feet in Asita's hair and Asita examined the birthmarks. Suddhodana held a naming ceremony on the fifth day, and invited eight brahmin scholars to read the future. All gave a dual prediction that the baby would either become a great king or a great holy man. Kaundinya (Pali: Kondañña), the youngest, and later to be the first arahant other than the Buddha, was reputed to be the only one who unequivocally predicted that Siddhartha would become a Buddha.

While later tradition and legend characterized Śuddhodana as a hereditary monarch, the descendant of the Solar Dynasty of Ikṣvāku (Pāli: Okkāka), many scholars think that Śuddhodana was the elected chief of a tribal confederacy.

Early texts suggest that Gautama was not familiar with the dominant religious teachings of his time until he left on his religious quest, which is said to have been motivated by existential concern for the human condition. At the time, many small city-states existed in Ancient India, called Janapadas. Republics and chiefdoms with diffused political power and limited social stratification, were not uncommon amongst them, and were referred to as gana-sanghas. The Buddha's community does not seem to have had a caste system. It was not a monarchy, and seems to have been structured either as an oligarchy, or as a form of republic. The more egalitarian gana-sangha form of government, as a political alternative to the strongly hierarchical kingdoms, may have influenced the development of the Shramana-type Jain and Buddhist sanghas, where monarchies tended toward Vedic Brahmanism.

Early life and marriage

Siddhartha was born in a royal Hindu Kshatriya family. He was brought up by his mother's younger sister, Maha Pajapati. By tradition, he is said to have been destined by birth to the life of a prince, and had three palaces (for seasonal occupation) built for him. Although more recent scholarship doubts this status, his father, said to be King Śuddhodana, wishing for his son to be a great king, is said to have shielded him from religious teachings and from knowledge of human suffering.

When he reached the age of 16, his father reputedly arranged his marriage to a cousin of the same age named Yaśodharā (Pāli: Yasodharā). According to the traditional account, she gave birth to a son, named Rāhula. Siddhartha is then said to have spent 29 years as a prince in Kapilavastu. Although his father ensured that Siddhartha was provided with everything he could want or need, Buddhist scriptures say that the future Buddha felt that material wealth was not life's ultimate goal.

So, if it makes you happy to claim Nepal as Buddha's birthplace, please do so.  And if anyone from India also wishes to claim that Buddha was born in India, also feel free to do so.  His father was Hindu, by the way.  To me, it is totally irrelevant where a fellow is born.  What he accomplished during his lifetime instead draws my attention and interest.