Hi Fingolfin, the FSM approach is quite a different implementation compared to the embedded (single/double) tap.
The main difference is that the embedded function uses the slope between two consecutive acceleration samples to detect tap events, while the FSM implementation uses the accelerometer raw data directly (without applying any kind of filtering). Moreover, the two implementation might also differs in terms of thresholds, ODRs, timing, etc…
There are at least a couple of advantages using FSM: in this way, you operates directly on the raw data (e.g. no needs to implement internal filters). Moreover, you can implement up to 16 FSM for tap recognition (different threshold, etc). Also note that the minimum recommended axl ODR for the is 417Hz, while the FSM can work also at lower ODRs (e.g. 26Hz), saving lot of power consumption.
Let me know if you need any examples.
