While there are technological and ideological differences between other blockchain DNS projects and Handshake there is one key difference that all alternatives share - they are orphaned TLDs. Handshake upgrades the current DNS system as a soft fork, where other projects create their own roots and TLDs that are inaccessible from current DNS system.
The first attempt at DNS on blockchains and was directly contributed to by Satoshi Nakamoto. BitDNS eventually became Namecoin.
Native TLD: .bit
Native TLD: .eth
ENS is a smart contract system on the Ethereum blockchain. Their main goal is to add web3 functionality to domain names such as sending cryptocurrencies to domains and pointing to websites hosted on decentralized storage services. While ENS technically has it's own root zone, they have decided to integrate the traditional ICANN root zone into their project allowing TLDs like .lux, .club, and .art to utilize ENS functionality. This means that Handshake and ENS are also compatible and complementary technologies for decentralized domain names all the way down the DNS stack from root zone to registrars to record management. .badass/ is an example of using an HNS domain on ENS smart contracts and XNHNS is an automated bridge for using ENS with HNS domains. The ENS team generally does not support HNS because HNS is prone to name collisions with ICANN root zone, violating their "responsible citizenship" principles.
ENS has a native TLD .eth that they created. The ENS .eth conflicts with the ICANN reserved TLD .eth for Ethiopia. Because of this ENS .eth domains will never be resolvable over DNS and therefore requires custom integrations for every part of the DNS stack whereas Handshake works by default with everything if your DNS is pointed to a HNS resolver. The Handshake blockchain reserves .eth TLD for the ENS team which resolves this issue and the ENS team has already claimed their TLD and HNS airdrop.
Another distinction between ENS and HNS which might change in the future is that ENS only offers SLDs and they are rented on a yearly basis with payments going to their multisig wallet. In contrast, HNS allows you to own TLDs with a single one-time payment and your HNS tokens are burned instead of going to a single entity.
The main difference between HNS and ENS is ideological more than technological. Handshake fundamentally changes the ownership model and power structures embedded in DNS, which is one of the most critical parts of the internet, by decentralizing the root zone. The ENS team is supporting corporate ownership, extractive fees, and censorship in the current DNS system by supporting ICANN over HNS.
Native TLD: .crypto, .zil