Mother's Day was invented by a little old lady (Ann Jarvis, and her daughter Anna Jarvis, the holiday was later repudiated by the younger Jarvis*), and later co-opted by Hallmark as (as far as I know) the first Designer holiday, for the purpose of selling cards. If you didn't know I was a cynic, now you know.
But at least, everyone knows who their mother is; there are usually witnesses. Of course, a person could have been switched at birth, and then you wouldn't know anything. But even in the best of all possible situations, it's possible that the identity of the father of somebody could be in doubt.
I think that, given that the relationships between a person and his or her mother, and his or her father are not symmetric, the two 'feasts', if you would, are not symmetric either.
A mother usually has a strong influence on a person's upbringing for various reasons; the more conservative or traditional the childhood of the person, the bigger the influence.
The influence of a father on a child varies far more greatly from one family to the next; some fathers are heavily involved with child-raising, while other fathers are determinedly hands-off.
In the past, fathers were more interested with the upbringing of their sons than of their daughters; they would supervise the boy doing all sorts of 'guy things,' while the mothers would model various 'girl things' for their daughters, again, in a traditional home.
But in the sixties and later, public schools took over the obligation of training all kids in both arts considered appropriate to women and to men. For instance, my own daughter learned woodworking and metalworking, in addition to needlework and cooking. Unfortunately, I think some conservative families were apparently unhappy with this, and I'm not sure that this plan has been continued. The school even taught all the children—regardless of gender—to operate a motor vehicle, and I believe that's standard across the country. Otherwise, it would fall to a parent to pass on this skill. (I don't know why I brought up that fact.)
So when we celebrate fatherhood and motherhood, we could be celebrating very different things. But it is not uncommon for fathers to feel that Fathers' Day is rather a consolation prize!
Arch
* P.S. The story of the founding of Mothers' Day is a long, and ultimately unsatisfying story you can look up.
Initially, Ann Jarvis began to work for what she called A Day of Peace for Mothers, as a response to the American Civil War that was just winding down. She felt that a large international congregation of women—mothers, specifically—might succeed in preventing violence, where men had failed.
Julia Ward Howe (a well-known poet and hymnodist) took up this effort, and briefly succeeded in establishing a Mothers' Day in June.
The movement to establish Mothers' Day in the present form, to remember and recognize the mother's of individuals, was taken up by Ann Jarvis, the daughter of Anna. Over the years, the feast was enjoyed more by florists and greeting card manufacturers than by mothers, a situation recognized by Anna Jarvis with increasing anger, and she set out to call for the abolition of the feast of Mothers' Day, with no success. She is said to have died penniless, and it is reported that she was buried with financial support from the florists and / or card manufacturers. So the popularity of Mothers' Day never let up, despite all Anna Jarvis' efforts to destroy it.