I don't think there is any discussion about 'soul' in Buddhism. So I don't agree that 'the Buddha says we have no soul'. I think that is a bad intepretation resulting from a cultural clash. 'Soul' is actually a very specific term to the Western religious tradition. (I think it is derived from the Jewish term for 'breath'). I personally believe in the reality of soul and think it signifies an aspect of the being beyond the purely personal. It is something rather like character, but somewhat deeper, and particularly connected to the qualities of empathy and beauty in life. Maybe it is not even an entity but a quality, 'soulfulness' - however I am sure it is real. But it is junior to Spirit which is the light of the divine within us (i.e. Eckhardt) and the 'ultimate principle' or ground of being (i.e. Tillich). (This is not Buddhist teaching but it is the view of many of the perennial philosophers and I endorse it.)
The question of the meaning of the teaching of no-self is often vexed. There is no complete unanimity in the Buddhist tradition about this topic. My interpretation of the teaching is 'every perceivable phenomenon is not self'. This does not actually say there is no self. It says that every perception or experience is not self (anatta).
So every arising perception or sensation is not-self (and also impermanent, and 'dukkha' - these are the 'three marks' of experience). But it should be interpreted carefully. It is not really an intellectual or ontological theory in the sense that Western philosophy would like. It is more an observation about the nature of our experience, or a 'critique of experience per se'. And it is something you can learn to see. The purpose of Buddhist meditation is to see this directly. You do notice that when you have learned this skill, you are much less prone to being 'hijacked by emotions' because you recognise them instead of just letting them take over.
I think the aim of the non-self teaching is to allow us to overcome, or see through, the reflexive identification we all have with passing phenomena, our entanglement with everything. Instinctively we are always clinging to things, experience and sensation without being aware that we are doing that. This is the 'ignorance' being referred to. We are deeply habituated without being aware. So we wonder why we suffer when things don't turn out or aren't pleasing to us - but deep down, we are responsible, because of the way we have 'constructed' (=vikalpa) our sense of reality. We cling, and then things get taken away. This is what is meant by attachment. Our sense of 'who we are' is deeply embedded in our situation, and while everything is going along OK, it is satisfactory, but then, as the teaching has it, everyone is prone to loosing what they like, getting what they don't like, suffering, illness, old age and death. That is the human situation.
The Buddha does not deny the reality of worldly happiness, of the ordinary joys of life, of making a living
and raising a family. But there is something beyond all of that, too, something much greater to aspire to.
So spiritually, this is really not so distant from many aspects of the Christian spiritual tradition, although I agree that it is a mistake to equate 'nibanna' with 'the kingdom of heaven', and of course the religious outlook of Christianity and Buddhism are quite different. (But there are intriguing parallels too, and some very interesting inter-faith dialogs and teachings, like an entire sub-tradition of Catholic Zen. Check out Ama Samy
and Ruben Habito
, both Jesuits and Zen Masters.)
I think the Eastern faiths and Christianity, however, are one in saying that egotism, selfishness, me first, or call it what you will, are the source of most of the suffering in the world. We have a deep expectation about the way things are, based on our notion of ourselves.
As for the 'sin' vs 'ignorance' distinction, I actually believe (and this is most unfashionable) that there is such a thing as sin, in fact I agree with some churchmen that the greatest trick of the devil was to convince the world that it doesn't exist any more. But at the same time, the concept of sin is not, in the parlance of modern self-awareness, particularly empowering. The Dhammapada: 'by oneself one is purified; by oneself one is defiled'. This is a very characteristic Buddhist teaching.
Perhaps the idea of sin is a trap, because it makes it like a malignant external force, whereas the origin of it, if you look very closely, might actually be inattention or lack of self-awareness. But I know it is a very difficult and sensitive issue, and would not like to be in the least prescriptive about it. Certainly some people benefit from a sense of being helped. It depends a lot on the type of person. At the end of the day, however, in Buddhism, the agency of real change in life is clear and direct awareness. This is in line with Christ's teaching 'Ye shall know the truth, and the truth shall set you free'. I don't think there is a sense of the Buddha 'taking on your karmic burden'. His last words were, according to tradition, 'all manifest reality is subject to decay. Work out your own salvation with diligence'.
I hope that is helpful.