What do you like best?
Pagerduty allows you to divide people into separate groups, so that if the severity incident is something related to a specific group, it is not necessary to trigger it for everyone. It also allows for escalation and "on-call" duties per week - this is extremely useful, as nowadays our helpdesk team does not even need to call us personally, since Pagerduty handles everything on its own and always knows who's on call.
What do you dislike?
I don't think this is entirely related to Pager Duty in itself, but to our own current infrastructure. The alert emails seem to be sent out to everyone (included the pager duty account, which triggers pager duty in itself). Thus, any subsequent email in the email chain may end up creating a different alert for pager duty.
Recommendations to others considering the product
Set everything up correctly from the start! Consider the type of alerts you need. For myself, it is extremely useful to set ALL alerts (SMS, Calls, Email), as it may be easier for me to look at my phone if everything's going haywire.
What business problems are you solving with the product? What benefits have you realized?
Our company provides a high availability solution for our customer. Pager Duty allows us to get all necessary hands as quickly as possible whenever an issue arises. This has also leveraged our helpdesk team's need to call development, DBA, etc, on their own, as they may sometimes not know entirely who's on call. Triggering an alert on pager duty is more than enough.
As a member of the dev group, if you get an alert and you are on call but not available, it is easy to escalate to the next dev on the line - without having to personally call them.