FOX-1 vs FOX-3:
The FOX-1, such as the sparrow, requires an STT lock for the missile to guide to the target. The seeker in the sparrow looks for radar reflections produced by your radar reflecting off the locked target. A TWS track does not provide sufficient guidance information to the sparrow, and will not track the tracked target; an STT lock is required for sparrows. Sparrows do not use datalink; when a lock is broken, it is often difficult to lock the target in time and continue missile guidance.
The FOX-3, such as the AMRAAM or phoenix, use datalink to guide to their targets, until they got close enough when their onboard radar turns on, locks the target, and begins guidance on its own. This is called “pitbull.” When tracking a target with either STT or TWS, the radar feeds guidance information via datalink to the FOX-3. In other words, the radar doesn’t require an STT lock for the FOX-3 link it does for the FOX-1. The FOX-1 looks for radar reflections, which is why STT is required. FOX-3s don’t look for radar reflections, as they get their guidance information from datalink, not the radar itself. The radar provides information to the datalink, not the missile. This is why FOX-3s can use TWS, and FOX-1s cannot.