The Thread border router carries network traffic
It connects the low-power Thread mesh to the wider IP network. If a device says Matter over Thread, the home needs a compatible Thread border router somewhere in the ecosystem path. Wi-Fi Matter devices such as a compatible smart plug do not need Thread for their radio path.
The Matter controller manages the accessory
It commissions devices, applies the ecosystem's permissions, and exposes controls and automations. Phones can assist setup without always serving as the always-home controller.
One hub can provide both roles
Some Apple, Google, Amazon, SmartThings, Aqara, and Home Assistant hardware combines the jobs. Check the exact generation and current software rather than the brand alone.
Write both requirements before purchase
For a Matter-over-Thread device, name the primary Matter controller and at least one compatible Thread border router. This prevents the common assumption that either role automatically includes the other.