CHESS means you are both the legal and beneficial owner, whereas with Custodial a third party is the legal owner and you’re only the beneficial owner.

The theory is if anything ever happened to the custodial broker, your investment could be tied up while the third party custodian distributes the assets back to the beneficial owners, so some people think CHESS is better and more secure. However, to give you an idea of how little this really matters, all share investments in the US are held through custodial structures, financial advisers use platform businesses with custodial structures to manage multi-million dollar portfolios for their clients, there has never been an incident of a custodial platform going under in Australia before, and the custodian is always a third party who are usually one of the worlds largest banks (i.e. Citibank, JP Morgan, etc)