A storage system in one embodiment comprises a plurality of storage devices and a storage controller. The storage controller is configured to monitor a differential between a storage volume of the storage system and a previous snapshot generated for that storage volume, and responsive to the monitored differential satisfying one or more specified conditions, to generate a subsequent snapshot for the storage volume. For example, monitoring the differential between the storage volume of the storage system and the previous snapshot generated for that storage volume illustratively comprises maintaining a counter indicative of an amount of data in the storage volume that has been written since generation of the previous snapshot. In such an arrangement, generating the subsequent snapshot for the storage volume responsive to the monitored differential satisfying one or more specified conditions illustratively comprises generating the subsequent snapshot responsive to the counter exceeding a specified threshold.