Halogen heaters and quartz heaters use basically the same heating element to heat up however the difference between a halogen heater and a quartz heater is in the type of glass used over the heating element.
Halogen heaters usually use standard glass due to the lower temperatures while quartz heaters use quartz glass to enclose the element.
Quartz heaters also put out more instant electric heat than Halogen heaters do so they're similar to each other although they are a bit different.