The verdict
For a small team that sells software or online and has a developer, I'd pick Stripe. It is the safest first choice because most tools in your stack already integrate with it, and you pay per transaction with no plan fee.
If you run subscriptions with usage, credits, several plans or custom enterprise terms, and you want finance to change pricing without a developer, pick Chargebee. If you have only a handful of flat plans, Chargebee is more than you need.
For most readers of this site, Stripe wins. You can add Chargebee later if billing logic outgrows it.