In this case, the smart contract aspect seems more like effective marketing rather than a technical advantage. This business could be operated identically with a centralized system.
Yeah it's definitely been done more for marketing but Axa's CEO is on record as saying he thinks the blockchain and smart contracts will have a lot of valid use cases in the insurance industry.
I suppose in this case the nice thing with the smart contract (assuming it works as advertised) is it's simple and verifiable. You know you'll get an instant payout if publicly available data shows your flight was delayed > 2 hours (regardless of reason). There's no human in the loop and no signed contract with weasel words that gets AXA off the hook from paying.
Sounds like a very roundabout way of doing things. I guess AXA feeds flight data onto the blockchain, you need to trust their data, and somehow pre-register the flights you are going to take with the contract?
I haven't used it but according to their FAQ, the smart contract uses publicly available flight data from multiple government sources to determine if a flight is delayed. The payout is based solely on that and the decision is out of Axa's hands (their website specifically says: payout regardless of reason for delay, including alien attack).
You can sign-up for the cover up to 15 days before departure. Apparently you input your flight number and credit card details. If the flight ends up being delayed more than 2 hours, their smart contract triggers an automatic payout to your card, no questions asked.