What do you like best?
What amazed me was how simple and small some of the changes were when I needed to do something complex. Most code changes were 1-4 lines long, to create some custom feature that would normally be 100 lines of jQuery or Dojo code.
What do you dislike?
Knockout differs from some more comprehensive frameworks (like Angular or Ember) where everything is included. Knockout provides data-binding, but lacks features found in full frameworks like routing or UI controls. However, you can mix and match other libraries in your app that handle those features, and it integrates very well. You can mix in jQuery UI user controls, or a library like Sammy.js for routing, and let Knockout handle the data binding and keeping up with the state of your application. It depends on how much you want to get your hands dirty with mixing in different libraries.
The one issue I had was that support for more complicated examples and integration with other libraries was hard to find. I'd often find the answer somewhere on StackExchange, or buried in a 5 year old blog that was abandoned 3 years ago.
Recommendations to others considering the product
There are frameworks out there that do more (like AngularJS or Ember). But if you need to create a single page application quickly, and you don't want to experience the crazy learning curve that those frameworks require, try Knockout.js. It's easy to use, and you'll be amazed at what you can create within the first hour. You can be very productive very quickly with this library.
What business problems are you solving with the product? What benefits have you realized?
I was tasked with creating a property search website for a government organization. Once we had the data model for the project, creating the single page web application and adding all the special logic within was a breeze. When change requests came (and we received a lot of them), I was able to handle most of them with just a few lines of code. I was amazed at how quickly we could turn around change requests to the client.