What do you like best?
Out of the next generation cross platform platforms, NativeScript is unique in that it fully allows you access to the entire Native platform. So you have access to the full development libraries as if you were developing a native iOS or a native Android app from inside your code directly. If also has a cross platform Declarative UI which allows you to make your UI once and use it easily on all platforms supported.
You can easily design a cross platform application in way less time than you would if you were attempting to create two separate apps using ObjC/Switft and Android Java. This allows you to have one code base and quickly build out an application meeting the business requirements.
What do you dislike?
At the current point in time, multi-threading can be a problem. Some things are by design already multi-threading. But somethings won't be multi threaded until later in the year (2016).
Tooling could be better, but it has been steadily improving.
Recommendations to others considering the product
You do need to learn some command line tools right now if you want full control. But they are very simple to use and you will be much happier using everything on your own box rather than the cloud.
What business problems are you solving with the product? What benefits have you realized?
Building a cross platform applications for clients. I can quickly prototype and app for the client, then rapidly iterate and get a app working exactly like the customer wants. Then from their I can make minor changes (if any) to make it work properly on all the platforms.