The alarm output on single loop controllers typically trips at the alarm setpoint and resets at whatever hysteresis value the controller uses.
The control output relay usually does NOT trip at setpoint - the setpoint is midway between the off-state and the on-state (the on-off deadband). So when looking at a temperature chart the setpoint is in the middle of the overshoot peaks and undershoot peaks.
But alarm relays change state (trip) at exactly their setpoint.
Because the alarms will reset when the temperature changes direction, if you need latching, then some controller models will latch an alarm relay; for those that do not you’ll need to drive an external, interposing relay and latch it and provide an unlatch circuit (switch).