It rests on what are the 'soul', the 'self' or 'this which is'. You can think of the difference this way:
- Religions believing in a soul (Hinduism, Christianity etc.) see the soul as representing the true you. The soul is a rock: solid, unmoving, unbreakable, eternal. Your good deeds will polish that rock to be smoothe and shine like gold. Your bad deeds with make the rock mould, decay, become rough. Once you die, this rock is judged (Abrahamic religions) or enters a new body as it is, to hopefully start being polished and eventually becomes a surreal eternal part of its surroundings (Hinduism).
- Religions without a soul (Buddhism's the only one I think) say the self is not a rock at all. The self is like a flame. You can think of the flame as being a single 'thing', but it's actually not, because the flame that burns before you sleep and still burns when you wake up is still there but it is not same flame. A rock will still be there unchanged, but a flame appears to be there but is never the same at any moment in time. The flam can transfer to a new candle and the old one is snuffed out, but is this the same flame? No.
So what is the flame? The flame is made up of several different elements that combine to make the flame exist. Yet they don't combine together to become one, like a rock; they're seperate things linked together appearing as one (like 'that flame') but in fact there is no 'one' in existence at all. Many ask "who created the world" but Buddha said the word "who" is gramatically correct yet entirely wrong; the question should be "what created the world". When you think what created the world, the question of "who" becomes irrelevant, blocks the mind, and doesn't reflect reality. Same kinda goes for saying "my soul" which is gramatically correct but completely wrong; it's more appropriate to say "this which is".
Looking at rebirth, right now the idea of being an invalid begger in a poor country is a horrific thought. However if you're born as one, you simply are what you are and deal with your situation. You yourself are existing right now with full consciousness ad have no idea how your consciousness got here, but you're dealing with it. You're the flame in that new candle, made up of the flame from the previous candle, yet you're completely different. This is an extremely free and liberating thought when you think of it. In a sense, you already are that invalid, and he already is who you are; you just don't realise it yet but one day it'll happen and you'll have no recall of how or why you're there, but you just are, and he'll be how you are.
Anyway, there are many different Buddhist sects so never take them at face value. Most if not all Buddhist societies have incorporated native superstition, native culture and an element of egotism of superior knowledge when writing of Buddhism. You don't even have to believe in rebirth to be a Buddhist, and some take rebirth and the 6 realms of the afterlife as metaphorical for states of mind in this lifetime only.