What do you like best?
Unlike applications such as Asana and Basecamp, which are largely just cloud based to do lists that work great for departments whose work load can actually be broken down into unrelated tasks, it's really geared towards developers who often work in a web of dependencies and with tasks that are part of a greater whole, that may be 2-3 layers deep in the hierarchy of that project's completion, that are rarely just completed or not, but instead have numerous statuses in between that make take days or weeks to transition through. In a lot of ways, Pivotal Tracker is just like Jira, but whereas Jira out of the box is basically a fully dynamic setup that forces you to architect out the connections you want and all the labels for your dev cycle, Pivotal Tracker comes out of the box as a template with the most commonly used features already set (your status options are : started, finished, deployed, accepted, your ticket types are: epic, feature, bug, chore). So unless you're hell bent on a fully customizable tool, Pivotal Tracker is great because you can get up and running right away, and will rarely find you need the changes to the existing setup anyways.
What do you dislike?
I wish bug tickets still allowed you to set an estimated time to complete. I handle most of the "bug hunting" at my job, and so it's hard for management to get an idea of where my time goes because of this unless I'm communicating to them constantly what I'm doing and when (i.e. how much time a specific bug is taking up). Which in turn makes it difficult for them to know when an edge case bug is worth continuing to pursue or if they just want me back on the main deployment for that month. I have a great boss, so it isn't as much of an issue, but if I were working elsewhere, I would be pretty upset this feature didn't exist.
I wish that when I created a view of "my tickets", that it still broke them apart with little headers showing what project dashboard view they were pulled from.
Recommendations to others considering the product
If you don't need a fully customizable project management tool, but do need a solid, dependable application geared towards development life cycles that will give you all the default options you need and has a quick spin up time, go with Pivotal Tracker.
What business problems are you solving with the product? What benefits have you realized?
Project planning, bug tracking, time management, deployment orchestration, employee time management. Using Pivotal Tracker has definitely helped us keep better track of who is doing what, where we're at with smaller tasks that a deployment depends on, making sure feature requests that have been backlogged eventually move up to the front of our workload and are completed, making sure the requirements the developers get match what the stakeholders are asking for, and making sure everyone is on the same page with knowing what needs to get done before the next release and can pick up work as they finish their current workload.. In turn, we've had a lot fewer instances of the ball being dropped, incorrect project requirements being worked into the code, and deployments going out with missing pieces.