I think they each have their place. More and more of what I do involves rich client work and so I often need Node just to build the client.
For API work, I think .NET is just getting better and better. Now that it's cross platform with Core, there are fewer impediments to choosing the right tool for the job.
With my projects it comes down to whether I use MVC for a more server-focused app or Node for one that's more client-focused. If you do use Node, you have the option of building universal/isomorphic apps. Of course, it mainly comes down to how you like to build your server side pieces, and which environment and ecosystem you prefer.
My current project is mostly MVC (we haven't moved to Core, yet) with a WebAPI piece, but I still use Node to build our rich client pieces. They're built with React and TypeScript. This allows us to continue building primarily a server-side app, while gradually migrating to more of a rich client approach.
The downside with Core is that it's still a work in progress. They've just released 1.1 but tooling is still not quite there. It might be closer when VS 15/2017 is released. One big piece is their move away from JSON project files back to a more flexible XML-based csproj model. They should be there soon. It's just indicative of growing pains.
The other big thing for me is EF7/Core, which is a rewrite of Entity Framework. There's a lot to look forward to here, but it's not at parity with EF6. If you're starting a new project, it's probably OK. If you have to migrate, it could present serious pain. It's the primary reason we haven't been able to migrate to Core, yet.
Regardless, if you used to use .NET and you haven't tried Core, you absolutely should. You don't even need VS. You can run VS Code and you can develop on Win/Mac/Linux/+ and you have the same choices for deployment. Also, as others have mentioned, their new Web server, Kestrel is also getting very fast.