What do you like best?
Twilio is a very extensive and comprehensive communications platform in the cloud. It has literally everything you need to start a company's inbound and outbound communications system from absolutely zero, or to revamp an existing local PBX system to use Cloud-based capabilities. What I absolutely love from Twilio is the enormous expansion possibilities that it has via programming (it supports Ruby, PHP, Perl and Java, among others), and for the non-initiated the "Getting Started" tutorials are just perfect to get a go in hours, not days.
What do you dislike?
There are so many useful and powerful features in Twilio that there is really not much to dislike; however, since everything has a margin for improvements, I'd add "Twilio Softphones", that can be deployed to endpoints (or remotely via email or other means) already preprogrammed with the required SIP domain authentication values. Currently, for SIP, each customer needs to provide their own softphonn software and program them by hand. That feature would be absolutely the best addition to top an already excellent communications suite!
Recommendations to others considering the product
Definitely go for it! Even if you internally use an on-premises PBX solution such as Asterisk or Cisco you can easily integrate it with Twilio via SIP Trunking and take advantage of the huge quantity of communication capabilities it provides! And it even provides you with a free Trial account (phone number included!) for you to experiment and evaluate it, so it's a no-lose scenario.
What business problems are you solving with the product? What benefits have you realized?
Our entire communications infrastructure is in Twilio. We previously had a mixture of several providers (both Cloud and on-premises) for that, but with Twilio we consolidated inbound and outbound telephony, SMS campaigns, IVR capabilities, call recording and customizable extensions, and even integration with our CRM platform.