Religion can be broken down into two aspects: beliefs and practices (traditions). The important aspect for the creation of a new religion is belief. While there are often variations within a religion there is usually one or two overarching beliefs which define it. For example, in Christianity the belief that Jesus of Nazareth was the Christ or in Islam the belief that Muhammed was God's prophet. Once that core belief is no longer true, a new religion is born.
For example: Judaism and Christianity both believe in Abraham and the idea of a chosen people who are beholden to a single omnipotent god. In the first century CE Christianity changed the belief of a single god to a godhead with the Christ being seen as part of God and no longer believing in the dietary restrictions. Once the core belief of one God alone was changed a new religion was born. This new belief created new traditions and practices like the eucharist, Good Friday and Easter.