With MQTT, the MT sensors regularly push readings (typically in batches) to the broker. Clients can subscribe to get the data from the broker.
If you have an application that supports MQTT, it's pretty much a ready to go solution. If you have non-Meraki IoT devices that also support MQTT, you have the benefit of everything going into a single platform. The data goes from the sensor, via the local MR/MV device(s) that the sensor connects to, direct to the broker.
With the API, you pull the sensor data on demand, either for 'now' or the readings over a period of time. Then probably store them somewhere for other applications to read the data. You are getting the data from Dashboard, not direct from the sensors.