This is a very common question to get confused over as they are similar ideas. It is important to remember that reincarnation is a Hindu belief while rebirth is a Buddhist one. both ideas are based on the religions' beliefs about the sould and the afterlife. They do not believe in an afterlife involving heaven or hell but in samsara. This is an endless cycle in which all humans are trapped, death is simply the beginning of the next life.
Hindus believe that each person has a permanent soul within them which is their true essence. When they die, this essence is transferred from one body to the next. This means it is still essentially the same person in the next life. In extremely rare cases, some Hindus claim to be able to remember their past lives.
Buddhists, on the other hand, believe that nothing is ever permanent (this idea is called anicca). Therefore they reject the idea of a permanent soul which moves from life to life. They say it is just their karmic merit (the good or bad results of their actions in this life) that transfers to the next life. The next life is somehow related but is not essentially the same as the last one. This idea is highly complicated and even many Buddhists struggle to get their heads around it!
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