Pivotal Tracker is meant for development teams and technical people - it's all about Scrum, velocity and story points. This really doesn't matter to your clients. Breeze has hidden away all that complexity and made a tool that is usable by ordinary people. Whether your are using Scrum, Kanban or Scrumban - it doesn't matter. Breeze just works - it's intuitive, easy to use, minimalistic and made for people not for machines.
Pivotal Tracker only allows you to assign story points to tasks - that's not very useful and helpful when you are dealing with clients. Breeze has an easy to use and understandable time tracking, just click on the play button and you start the timer. You can also log time manually and estimate your every task. The time input is super easy to use and accepts every kind of time format - 1h, 1hour 1h30m, 1:30 and more.
Pivotal Tracker really doesn't have any reporting functionality that is useful for sending out invoices or generating reports. Breeze shows you everything you and your teammates have done and how long it took. It shows you if you are over or under budget and how much time was spent on every task. To top it of you can send your reports directly to your clients in CSV format or better yet send it directly to Freshbooks to generate an invoice.
Breeze is an agile project management tool that also gives you a bigger picture. Pivotal Tracker doesn't have any calendar or resource management functionality. Breeze gives your every project a calendar where you can see all your tasks and get an overview when something starts and ends. Breeze also has a special resource view where you can plan your team's workload.