There are a number of similarities between the two religions; probably because Buddha himself was Hindu before his enlightenment. Both believe in reincarnation, life in physical form after death. Although this seems desirable, it means one must deal with the many hardships and sorrows of human existence all over again. One hopes to break this cycle and enter into a heavenly state. Both are religions of salvation; both believe that by breaking the cycle of reincarnation one enters into a state of eternal rest. From there, there are significant differences.
Hinduism believes that one is born into a caste, or social status based on ones previous life. Although the Indian government has outlawed the system, belief in it is widespread. Buddhists do not practice the caste system, rather they do believe that ones past life has some determination on ones future life; but it does not bind one into an ironclad social sect. They do believe that reincarnation continues until the cycle is broken. They also believe that all suffering is caused by desire; and one can only break the cycle of suffering by ending desire.
Hinduism does contain a large pantheon of gods; but it is not necessarily true to state that Buddhism has no Gods. There are variations of Buddhism, one of which, Mahayana, or the "greater vehicle, worship Buddha himself as a god. This sect is popular in Korea, Japan, and China.
Hindus believe that if they lead a balanced life of obedience to the moral and spiritual law, pursue honest prosperity and enjoy social and physical pleasure in moderation, they will achieve salvation and enter moksha, a state of heavenly bliss. Buddhists believe that when the cycle of reincarnation is finally broken, they enter a similar state known as nirvana.
One final but important difference. Buddhism is a missionary religion which actively seeks converts. This accounts for its spread throughout east Asia. Hinduism is not missionary, but is rather culturally defined. It is seldom found other than among persons of Indian ethnicity.
In general, Hinduism is more of a formal religion than Buddhism is. By that, I mean that Hinduism believes strong in gods and in rituals while Buddhism does not.
Hinduism, of course, has a pantheon of gods that are worshipped. By contrast, Buddhism does not really have gods. To the extent that Buddhism believes in gods, it sees them in a way similar to Catholic views of saints. That is, the gods are beings who are not themselves divine and perfect but who can help good humans on their way to enlightenment.
Because Hindus worship gods, they have a number of rites and rituals that they observe. By contrast, Buddhists do not believe in the efficacy of such things. To them, meditation, along with right action and thoughts in day to day life, is what is needed for salvation.
In these ways, Hinduism is more of what I think of as a formalized religion while Buddhism is more philosophical.
I have gone through the other answers and I am not going to touch those points.
Hinduism believes in god, its true but while writing all scriptures of the Hinduism like Vedas, Purana, Srimadbhagavatam, Srimadbhagawatgita etc by the sage named Krishna Dwaipayana Vyasa and his disciples like Pailya, jaimini and others divided man into four castes like -Brahmans, Kshatriya, Baisya and Sudra. He also divided the works to be performed by these castes. Brahmans are to worship, Kshatriya to rule, Baishyas to do business and Sudras to cultivate and look after welfare of other three branches. The Brahmans lost their sanctity and began to exploit other in the name of religious matter like the monks in Christianity. Condition of other branches deteriorated and suffered a lot.
Buddha, who was the prince of an Indian King named Sudhodhan of Kapilavastu, saw this and felt much for the poor condition of people. He also utterly realised how people became sad at the death of their near and dear one. He found out that it was because of ignorance. So he went out inquest of real knowledge leaving behind his legacy and his wife.
He found that God is the mass of energy and it is difficult for all to realize without knowledge. So he taught-a) Dharmam saranam Gaschami b)Budham(knowledge) saranam Gaschami c) Sangham saranam Gaschami. He appealed people to come unto him, acquire real knowledge without which people suffer from weal and woe and to acquire them together in unity. He taught people how to get 'Nirban' to be free from all anxieties by acquiring knowledge that this phenomenal world is only a perception nor real. Then would aspire nothing and aspirations are the real cause of our sadness.
Buddhism is more of a way of thinking, based around a non-deity figure while Hinduism is a history of interactions with gods and men as well as their teachings.
Hinduism teaches one how to worship gods, while Buddhism teaches one the path to Enlightenment; a self-realization.