Whom should child love, respect, and care for more and why? Or, should he equally love both parents?
In the section of the 10 commandments, it mentions the father first "honour your father and your mother" with the verb of Kavod - honour.
However in parshat Kedoshim (Leviticus, 19:3), it mentions the mother first where the verb is Tir'u from the root Yirah - fear.
On the second of these, Rashi states:
[Every man] shall fear his mother and his father: Here, Scripture mentions the mother before the father, because He is privy to the fact that a child fears his father more than his mother [and therefore, by mentioning the mother first, Scripture emphasizes the duty of fearing her also. However,] in the case of honoring [one’s parents], Scripture mentions the father before the mother, because He is privy to the fact that a child honors his mother more than his father, since she wins his favor by [speaking kind and loving] words. [Therefore, by mentioning the father first in the context of honor, Scripture emphasizes the duty of honoring him also]. — [Kid. 31a]
Other laws, e.g. death for cursing or striking a parent, applies equally to the father and the monther. Similarly the laws of mourning apply equally for a mother as to a father.
Therefore they are treated equally in halacha.
There are some roles however that are specific to the father. If the child is a boy, it is the responsibility of the father, not the mother, to arrange Bris Milah, and if he's a first born then Pidyon Haben, and also the responsibility of the father to arrange the son's education, both Torah and Parnasa, and to find him a bride. (Gemara: Kiddushin, around Amud 29)
That doesn't make him more "important". They are just his roles.