A terrarium is usually a sealable glass container containing soil and plants, and can be opened for maintenance to access the plants inside. However, a terrarium can also be open to the atmosphere rather than being sealed depending on the plants you use.
A planter is a flower pot or box for plants. It can comes in all kinds of materials.
Succulents are plants that store water in their stems, roots, and leaves. There are about 60 different plant families within the group of succulents. Cacti are fleshy plants that store water, making them part of this group. Therefore, all cacti are succulents.
Cacti are simply a family, or sub-category within the group of plants collectively known as succulents. They range from tall and thin to short and round, and they usually do not have leaves or branches. In order for a succulent plant to be considered a cactus, the plant must have areoles. Areoles are small, round, cushion-like mounds of flesh where spines, hair, leaves, flowers, and more grow from the cactus. Areoles are only present on cacti, not all succulents.
Take a good read of this article by World of Succulents. While the article gives you a detailed information about the plants, do take note that there will be a slight difference in terms of caring for them because of climate differences.