It depends where is the target timeslot : on the same cell (intra cell ho) or on a neighbour cell (inter cell ho).
Assignment Command is used for INTRA-cell ho, while the Handover Command is used for INTER-cell ho.
Intra cell ho is always “synchronous”.
Inter cell ho is mostly “asynchronous”, unless both serving and target cell are synchronized (sharing a clock). In such case, the HO could be “synchronous”, meaning that there is no need for MS & BTS to re-synchronize after the HO (steps which are done at the very beginning of the HO “welcoming” procedure in the target cell, right after the HO ACCESS).