الأمي = the my mother (its not correct at all)
Here, the speaker is telling or informing us that his mother IS important. Hence أمي مهمة
If it is (my important mother), here you are not informing, you are simply attaching the adjective (important) to the noun (mother) which is called "attributive adjective". The two sentence serve different purposes even in English. My important mother is: أمي المهمة - notice how the definite article (AL) is attached to the adjective. The attributive adjective follows the noun it is attached to in gender, number, and definition. Here we attached AL to the adjective because the noun is defined (by the virtue of the possessive suffix ـي meaning "my"), and also the noun is original a feminine noun, that the adjective comes in a feminine form AND defined: المهمة
While in the original sentence, the adjective followed the gender only but not definition because it is predicative.
Reflect on German: Der Wind ist kalt and Der kalt Wind or kalter Wind. Same situation and exchanged relations between adjective and noun, informative (predicative) and attributive.