The verdict
If you run an agency, consultancy or similar firm and invoice for work done, pick Productive. The questions it answers are the weekly ones: are we profitable on this client, and who has capacity.
If you run a software or subscription product with complex pricing, usage charges or several plans, pick Chargebee. Productive is not built to bill recurring plans.
Some businesses sell both a product and services. In that case the two can sit side by side, and the real question is which problem hurts more today. Do not buy either one to solve the other's job.