Damage to the cerebellum would likely lead to problems with memories while damage to the hippocampus would lead to problems with memories.

Damage to the cerebellum would likely lead to problems with procedural memories while damage to the hippocampus would lead to problems with long memories.

Explanation:

Procedural memory refer to the unconscious memory that allow us to do tasks. This is the memory about how to do thing, skills, abilities, how to use some objects, or how to move the body. All these kind of memories are to do procedures, and the cerebellum is the center of that. The cerebellum also has big part in the in balance, motor control, language, or even emotional functions.

On the other hand, the hippocampus is essential in memory functions, specifically, the transfer from short memory to long memory. Also, the hippocampus control the spatial memory, and it has the ability of growing new neurones. Nowadays, there are plenty researches about the hippocampus, because most of it is a mystery still. However, the most important function found that this organ perform, it's about memory, according to researches about people with Alzheimer, the first part of the brain damaged is the hippocampus, as an effect, they cannot remember long memories.