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They both do produce a controlled voltage. A bandgap is a circuit (usually with temperature compensation) that is mostly used a constant stable reference voltage.
A regulator (for example a linear regulator) is a circuit which uses a series transistor as a variable resitor to regulate the voltage across a load. In order to provide a constant output a feedback loop is established usually comparing the output voltage (or a portion of the output voltage, using a resistive divider) to a known reference voltage (like a bandgap reference from above). So a bandgap is an integral part of the regulator system.