Common and mutual are close in meaning, but they are not the same.
Use common for something shared by members of a group, or something usual and widespread. It does not require a two way relationship. For example, We share a common goal means several people have the same goal.
Use mutual for something that goes in both directions between people, groups, or sides. It suggests reciprocity. For example, Their respect was mutual means each person respected the other.
- Common: shared by all, general, usual
- Mutual: shared equally between sides, reciprocal
A helpful contrast is this: a class can have a common interest, but two friends can have mutual trust. If the idea moves both ways, mutual is usually the better choice. If it simply belongs to everyone in a group, common is usually right.