Which is the better way to measure the performance of an online marketing campaign for a mobile app, the conversion rate or cohort retention?

Neither.

The only way to measure performance, from a business point of view, is SALES, $$$, or leads.

While conversion rate and cohort retention are valid statistics and can be used for other purposes, they cannot be deposited in the bank or used to pay the bills.

Even adoption of an app by millions of people can’t be deposited. But the revenue from ads shown to those people certainly can!

There are many worthless statistics and metrics which can be gathered. But it’s like a car. You can measure the optimum fuel pressure, the foot pounds of torque generated by the engine, the efficiency of the transmission- but none of those will get you to your destination when you want a coffee!

Gather and use metrics- just understand their limitations and you will go far!