Both Hinduism and Buddhism have been originated from India. They have a long history. Hinduism is believed to be originated way back some 5000 years ago, while Buddhism came in much later. While the Hindu scholars go with the ancient texts like the Vedas and Upanishads, they are a firm believer of the supreme trinity i.e. the creator, the preserver and the destroyer; also the concept of reincarnation holds an affirmative position in all Hindu teachings. On the other hand Buddhism is based on the teachings and experiences of Buddha. Its origin was observed after the enlightenment achieved by Lord Buddha. Buddhism does not follow any Hindu scriptures and do not believe in Atman or the Brahman, rather it preaches the attainment of no-self stage as the ultimate aim of knowing the truth. While we find that both of these religions believe in Karma, but may perceive it differently. Buddhism does not believe in stages of life, while the Hindus believe in four stages of life called the ‘ashramas’. The Hindus consider salvation or ‘Moksha’ as the ultimate goal of any human being alive in this world; Buddhists believe in detaching from the self and thus freeing oneself from all the sufferings of the world as the main goal for all humans. It is clearly observed that both these may share the same vision but the ideals and practices to get to that vision are different. While Hinduism advocates the self as the supreme, the Buddhist believe in a selfless approach to attain true happiness.
Comparing the Atman and the no-self theory
Atman, in Hindu philosophy literally means one’s true self. It may be explained as an immortal aspect of any living being. Atman, is eternal or spiritual in nature, it travels from one body to other after one’s death. The basic aim of all Hindu teachings is to break free from worldly pleasures and return to the Brahman. Hindu philosophy believes in reincarnation, which is all about wandering of the soul from one body to another. One may attain the highest level of wisdom i.e. the Atman, through meditation and yogic practices.
What is Atman? How does it look? What are its dimensions? , all these questions are not answerable until now and will not be answered ever. 
Atman can only be experienced. It is present in all humans but is embodied by the physical and sensory world. While the Atman is beyond our mind and senses, it does share all our experiences and doings. We are bound by external world for our deeds but to attain Atman one needs to achieve a state of stillness both from the body and the mind. It is widely believed that senses are ruled by our mind and the mind by our intellect and above the intellect is the ego. The ego holds a key position in all the beings. Ego is nothing but merely a self created illusion which controls our actions. To attain Atman is to move away from our ego, one who rise above ‘the me’ will realize the real ‘self’.
Atman as mentioned in the Upanishads cannot be reached by someone who is a scholar or a learned person, but by someone who commits to know the truth. Thus we find that Hindu theory of Atman deals with the realization of one’s true inner self, and this can be accomplished only when the mind and the senses does not interfere in the process of soul searching. Hence going by the Hindu teachings our Atman is the real ‘self’.
On the other hand from the Buddhist view there exist no soul which is permanent in nature.  According to Buddha, nothing is permanent or fixed every living thing is subject to change from the time of its formation to the time of its destruction. Buddhism believes that there is no self, not for any person or for any being; the universe is empty of self. This ego or self creates a sense of attachment which further restricts us from attaining or knowing the truth. Every living creature keeps changing and so does its reasons for existence. Early Buddhists thus believed that a human should not be known by his/her name or form; rather he should understand the process of constant change and be aware of ‘becoming’. This will keep him away from all the sorrows and sufferings of life, because if he, who recognizes that everything will change, will not experience grief in times of tragedy and will not be overwhelmed with what he gains. While Hinduism and many other religions support the theory of self awareness as the ultimate aim, Buddhism is in favor of nirvana or non existence. It is well described in the Buddhist literature that the continuously evolving nature of things around us may cause a lot of suffering and to get rid of the pain caused we should go in for closure and shift into a state of non existence rather going in for self awareness or soul searching.
The Buddhist view of self is quiet different from the Hindu self. In Hinduism the self or the Atman is indestructible and is the highest, while in case of Buddhism any self is an ordinary self which experience pain, happiness and other emotions but is not permanent. This soul may move from one body to another but still is open to all kinds of illusions and experiences. Thus Buddha preached ‘anatta’. Anatta means non self, a state of emptiness where you are devoid of everything, may it be a person or a specific thing. In this state you are visible to all and to your own self. A non-self state may sound very insecure but is the only way to attain the truth.
No-self theory: The Essence of Buddhism
All Buddha’s teachings are completely based on the concept of ‘anatta’ or ‘no-self’. This idea of non-self is thus an integral part of Buddhism. Buddha defined ‘self’ as an illusion; it is this illusion which produces all the wrongs and evil in the world. Truth can only be understood by one who accepts the concept of no-self and emptiness. Peace can be experienced if and only if there is no ‘self’ or ‘ego’.
In order to understand a no-self state let us define what ‘self’ means. Self means ‘me’; we may be identified by our name, our age, and our occupation and also by our relations with other humans. But is this self the real me or is just a matter of speech. The self is controlled by the ego; ego is nothing but a desire to be i.e. to be a mother, to be a doctor or to be happy or to be rich. This craving leads to attachment and attachment to persons, things will never make way for truth. Attachment in turn will result in clinging; we would want to cling to people, to things, to situations because these will provide us our identity. To attain anatta or a no self is to get rid of all the clinging and be free of the ego. If this sticking and ego goes, with it will go all the sufferings. Thus a process of liberating oneself from all kinds of worldly attachments and clings will consequence in an empty state and a state of nirvana or no-self. A non self state may not necessarily want you to leave your family or work, but will open up your mind to a clear understanding of the world around you. This no-self theory given by Buddha aims to relieve people from the misery they encounter due to sufferings and pain. This theory may bring about peace and harmony in the world. According to this theory every human being should indulge in detachment to reach the ultimate goal of truth. Meditation is an important aspect of all Buddhist teachings and plays a vital role in clearing up the mind from all that is not required. It is also found that though this theory is readily accepted by most Buddhist, it is still not clear as to when this was preached by Buddha, it also presents some contradictions with the theory of rebirth which is accepted by Buddhists.
According to me personally, the theory of no-self is much more justified. The theory can be understood by a simple example; a person who purchased an expensive article or a jewel sees it as an investment for the future. He puts it in his safe with all the locks. But he is always worried about the same to be stolen, if he goes out on a vacation or a holiday for his relaxation and happiness for a few days. And one day if it did get stolen all he is left with is nothing. He will now be happier and much more relaxed because he has nothing to get worried about now. Thus in this case emptiness proved to be the key to happiness and a deeper understanding of this concept may lead a path to truth and enlightenment. This theory teaches us to live a life of detachment. If we are devoid of all clings of the world, we would neither feel the pain nor sorrow. But in order to attain such a state of non self, one must be truly committed to know the truth. A person should be courageous enough to understand and practice this theory. Meditation can prove to be of great help as suggested by all the Buddhist teachings. Meditating is a state where in, one is devoid of his surroundings and concentrates on his mind and body. With continuous practice of meditation we can possess clearer mind which can understand the importance of detachment. And this realization may bring positive change in our lives. Hinduism also supports and advocates the importance of meditation in attaining salvation. Meditation nowadays is considered as the best stress buster, it helps you to gain control on your senses and thus allow us to open our minds to the real self in all of us. Thus to gain best of both the school of thoughts i.e. Hinduism and Buddhism, one should start with a simple step of meditating and further explore our inner strengths and weaknesses. To know the real ‘me’ would be the best outcome of all the teachings.
After going through enough resources and texts which deals with the insights of both Hinduism and Buddhism, it can be concluded that both the religions are quiet similar in some thoughts but perceive every component of being with different point of views. While Hinduism is not just a single religion or founded by a single saint or prophet, it seems to be an amalgamation of various thoughts and religions which were developing and some already existed in the early India. In case of Buddhism it may not be a religion but a group of thoughts and experiences which were gained by Gautam Buddha during his lifetime and change in the thinking process of developing India which was further accepted by people as a religion all over the world. In case of Hinduism idol worship is considered as an important element which leads to moksha, on the contrary Buddhism does not support the worship of any kinds of images or idols. Both the religions fully agrees on giving up one’s self to gain individual salvation either by better understanding of ones true self or by attaining the state of nirvana. Both of these religions may contradict each other at some point or the other, but are being well accepted and practiced by people in harmony not only in India but world over.
Thus both the religions teach us to focus on the truth and not on the worldly pleasures for ultimate happiness and peace of mind. Both support the theory of Karma i.e. ‘the law of consequence’ and that of rebirth. Both teach that, one who is born will have to die. A person may attain moksha or nirvana only if he has control over his mind and body.