On the surface, what Solomon requests and what Adam and Eve desire seem similar, but on closer inspection, they are very different. The words that are typed in bold in the question serve as a good starting point for examining these differences.
Whereas Solomon asks for the ability to “discern between good and evil,” Eve seeks the “knowledge of good and evil.” If we rearrange the words in the Genesis text, there are dark connotations inherent in the very idea of desiring the knowledge of evil.
But even if their objectives were the same, even if Eve was simply seeking wisdom, there is a stark contrast in their methods. Solomon makes his desire known to God. Adam and Eve, like thieves, act in secret and in willful disobedience of God’s commandment.
There are also clear differences in intent and purpose. Solomon asks for wisdom in order to be a more just ruler. Solomon’s intentions are pure, as is evident in God’s response: “It pleased the Lord that Solomon had asked this. God said to him, ‘Because you have asked this, and have not asked for yourself long life or riches, or for the life of your enemies, but have asked for yourself understanding to discern what is right.’” (1 King 10-11).
Eve, however, “saw that the tree was good for food, and that it was a delight to the eyes, and that the tree was to be desired to make one wise” (Gen 3:6). Hers was a desire for self-gratification and gain.
But perhaps the biggest disparity lies in their hearts. The text tells us that “Solomon loved the Lord” (1 King 3:3). Eve, on the other hand, is easily persuaded into thinking that God lied and selfishly withheld something that would be of benefit to men.
And finally, in contrast to how Solomon humbly refers to himself as a child and a servant before God (1 King 3:7-9), Eve is lured by the temptation to become like God (Gen 3:4). Pride or proudful ambition seems to have played a crucial role in the fall of both angels and men.
Having formed all the arguments and accusations against Eve, I feel a growing sympathy for her. In her, I see the reflection of my own faults and failures and those of mankind, of our human nature. After all, who among us can say that s/he is free of pride and immune to temptation? Even Solomon would come to betray the wisdom he had received and succumb to temptation in time.