What do you like best?
Velocity.js is probably the best standalone tool for handling animation and tweening in webpages. Lots of cool features and really easy to get running.
What do you dislike?
Most of these really cool features aren't practical for most situations. This is very fun for playing around, but with more sites moving toward static simplicity, this seems more like a fun weekend experiment for casual users.
Recommendations to others considering the product
This is a great product, but if you are already using jquery, be careful in including this as well. Make sure you need the additional features that come with the overhead of working with an additional library.
Also, make sure that you can't solve your problems using css instead, which is designed for animation and often leads to better and more performant results.
What business problems are you solving with the product? What benefits have you realized?
Making fun web pages and experimenting with various web technologies. It's good to keep up with different web frameworks in case you need them, and velocity js is likely the best full featured solution for web page animation. I'm sure it can be used for other things, including game animations, but there are usually other solutions for those as well with better integration that still get the job done.