Both are used to mitigate the effects of voltage dips. Dips are characterised by the depth - the retained voltage - and the duration. Short and deep dips are best served by a DVR while long and shallow dips are the province of the voltage regulator.
A voltage regulator has no energy store. It has a transformer secondary winding in series with the supply. When the input voltage moves outside the tolerance band the primary of that transformer uis driven to boost, or in anti-phase to reduce, the voltage appropriately. Because the load voltage is kept constant, the power to the load is constant so, when the input voltage falls, the input current increases. The current capability of the supply and the device itself limits the working range to about +/-30 % of nominal voltage.
A DVR has an energy store, so requires no additional input power (in the short term) to boost the voltage during a dip. A DVR can correct a dip to 0 % retained voltage. But the DVR has a limited energy store and so is suitable for short-term effects only - it cannot correct for long term under voltage, for example. Also, the store has to be recharged between events so it is not suitable multiple dips are expected frequently. Typically, DVRs use super capacitors, large secondary batteries or high-speed flywheels as energy stores.
Unsurprisingly, DVRs are more expensive than voltage regulators.