What's the difference between a bank and non-bank lender?

The short answer comes down to where they get their money and who regulates them.

Banks are authorised deposit-taking institutions (ADIs). They can accept deposits from customers and may fund home loans using a combination of customer deposits, wholesale and other funding sources. Because they hold the public's savings, they are regulated by the Australian Prudential Regulation Authority (APRA), to ensure their financial stability. They are also regulated by the Australian Securities and Investments Commission (ASIC) and the Australian Competition and Consumer Commission (ACCC), which focus heavily on consumer protection, fair trading, and responsible lending.
Non-bank lenders do not take savings deposits. They borrow money in bulk from large institutions or investors to lend out as home loans. They are not regulated by APRA, but are regulated by ASIC and the ACCC, which focus heavily on consumer protection, fair trading, and responsible lending.

How offset facilities work: Banks hold your offset money in a separate savings account. Non-bank lenders place your offset funds into a sub-account built into the loan itself. In practice, they both function in a similar way - lowering your interest while giving you access to your money - however with a non-bank loan the amount of money you can put into your offset is limited to the total balance of the loan.
What happens to the offset facility if the bank/non-bank lender goes under?: Banks are covered by the Government's Financial Claims Scheme (FCS), which protects savings deposits up to $250,000 per account holder if the bank fails. Non-bank lenders are not ADIs which means they cannot hold deposits and so funds held in a sub-account are not covered by this scheme.
What happens to the mortgage if the bank/non-bank lender goes under?: If a bank or non-bank lender goes under, your loan assets and the servicing rights would be sold or transferred to another financial institution. You won't lose your home, you cannot be forced to pay off the loan early (unless you are in default), and your repayment obligations remain in effect under your new lender.