What do you like best?
Using a push notification platform such as Urban Airship can save a great deal of time instead of having to build one in-house.
The ability to create channels and segments, schedule and batch notifications is extremely useful.
The setup (iOS) is pretty straightforward, although migration between major versions of their SDK hasn't always been seamless (but that is expected, specially when the updates reflect major changes in Apple's own APIs!).
Their support team was always helpful when required.
What do you dislike?
I used to use Urban Airship in some of my personal apps a few years ago, and being an iOS developer with little server-side experience, having a platform like this was a life saver.
However they unfortunately got rid of their free tier, and the lowest price plan became too expensive for an indie developer.
A lot of my clients still use it however, on larger projects with real budgets!
Their SDK seems pretty solid these days, but we have been faced with some pretty major issues in the past, notably because of the fact the 'detectProvisioningMode' feature was broken and resulted in all our production users being registered to the development account!
Recommendations to others considering the product
Although the feature seems to work correctly these days, I would suggest not using the 'detectProvisioningMode' and specifying the environment manually, either by code or using a separate config file for development and production environments!
What business problems are you solving with the product? What benefits have you realized?
Not having to build an in-house push notification server! That said, there are plenty of open source products which let you build a push notification server, though implementing things like channels, segments, scheduling and scaling is less trivial.