They are completely different. With information, in the context you gave, of can only indicate ownership/possession. It is otherwise incorrect.
Information of your family
Means information belonging to your family.
To say information relating to something, we can use about or on. For example:
My family has a history of diabetes
This is information on my family or about my family. We wouldn't (in most contexts) say it is information of my family.
It's not common to talk about information belonging to someone in this way, so when you use "information of [someone]", it should be very clear that the information belongs to them in a special way, for example, it is a private record with protected status (like medical records). It may be better to use a clearer way to indicate possession in this case, for example:
information belonging to my family
my family's information
Another context where information might be followed by of could be when we are describing the information itself, for example in a phrase like
information of a sensitive nature
This does not mean information about "sensitive nature", but describes the information as sensitive (so it might need to be kept private). Similarly:
information of this kind is considered sensitive
This means the type of information we are talking about (such as medical records) is sensitive.