Cacti and Succulents are indeed related and in the same group of plant family.
Cacti store water and are fleshy plants just like Succulents do so Succulents are really Cacti but just have a different name.
But for a Succulent to be considered a Cactus the Succulent Plant must have some areoles on them.
The Areoles are small, round, cushion ike mounds of flesh where spines, hair, leaves, flowers, and more grow from the Cactus or Succulent.