The verdict
If your team loses track of tasks, owners and deadlines, pick Asana. If you sell time and invoice by the hour, pick Clockify.
They are not rivals, and you can run both. Asana holds the tasks, Clockify holds the hours and your accounting tool receives the result. When I used Asana, it had no built-in time tracker, so we added a separate one. Check what your Asana plan offers today before you promise clients hours-based reporting. If you can only start with one, start with the problem that costs you more. For most small service businesses that bill by the hour, it is the missing hours.