Importar has two main meanings:
- Intransitive verb equivalent to matter
- Transitive verb equivalent to bring inside and import
As a transitive the construction admits the objeto directo. e.g.
mi madre importa productos europeos
The subject (mother) acts (imports) over something (European goods)
Your question, however, is about the usage when it is intransitive. The catch is that intransitive verbs (the same happens in English) do not allow objects. For example:
yo camino (I walk)
There is no thing I could act upon with the act of walking: I just walk.
So if we used the form
mi madre importa
It means that mother is important by herself. There is no one else involved.
In order to use importa to express that someone is important to someone else we need to add a pronoun from the following list:
me, te, nos, os, se, (le, les)
Thus we need to write:
Mi madre me importa mucho (she is very important to me)
Me importan mucho los diamantes (they are very important to me)
Referring to another person we could write:
A mi madre le importo mucho (I am very important to her)
Other examples:
¿Te importa si hablamos de otra cosa? (Does it matter/would you mind if we talked about another subject?)
No nos importa el lugar (the place is unimportant)
Finally, importante can be used as an adjective. e.g.:
Mi madre es importante para mi
Los diamantes son importantes
El lugar no es importante