There's Enigma "v1" (SGX only) and "v2" (SGX + MPC). My understanding is that in both schemes, all code will be public, although inputs and outputs can be private. So e.g. if a company wrote a multisig contract, the multisig parameters could hint at the identity of the company. Zexe seems better in that sense, since the code is private.
On the other hand, Enigma v2 could enable certain use cases that ZKPs don't. E.g. MPC protocols can solve the classic Millionaires' Problem, while ZKPs don't help there.
Most of the authors are affiliated with Zcash, so they're probably thinking about adding this as a feature of Zcash rather than a new system.
This doesn't use hardware enclaves, so it's a totally different approach to privacy/validation. I'm not sure about the trade-offs and how that impacts their respective niches.
