The trouble with J.S. Mill, Jeremy Bentham, and western liberalism generally, for him, were their progressive, optimistic interest in maximizing happiness (though he announced his own “formula for happiness: a Yes, a No, a straight line, a goal”) and pleasure and minimizing difficulty and pain.
To be more specific, what Nietzsche really had a problem with was the idea that pleasure and pain were clearly mutually exclusive. Alain de Botton wrote a nice passage explaining his antipathy toward utilitarianism:
Yet traces of his horticultural enthusiasm survived in his philosophy, for in certain passages, he proposed that we should look at our difficulties like gardeners. At their roots, plants can be odd and unpleasant, but a person with knowledge and faith in their potential will lead them to bear beautiful flowers and fruit — just as, in life, at root level, there may be difficult emotions and situations which can nevertheless result, through careful cultivation, in the greatest achievements and joys.One can dispose of one's drives like a gardener and, though few know it, cultivate the shoots of anger, pity, curiosity, vanity as productively and profitably as a beautiful fruit tree on a trellis.But most of us fail to recognize the debt we owe to these shoots of difficulty. We are liable to think that anxiety and envy have nothing legitimate to teach us and so remove them like emotional weeds. We believe, as Nietzsche put it, that 'the higher is not allowed to grow out of the lower, is not allowed to have grown at all...everything first-rate must be causa sui (the cause of itself).'Yet 'good and honored things' were, Nietzsche stressed, 'artfully related, knotted and crocheted to...wicked, apparently antithetical things'. 'Love and hate, gratitude and revenge, good nature and anger...belong together,' which does not mean that they have to be expressed together, but that a positive may be the result of a negative successfully gardened. Therefore:The emotions of hatred, envy, covetousness and lust for domination [are] life-conditioning emotions...which must fundamentally and essentially be present in the total economy of life.To cut out every negative root would simultaneously mean choking off positive elements that might arise from it further up the stem of the plant. We should not feel embarrassed by our difficulties, only by our failure to grow anything beautiful from them.
He thought Mill, Bentham, et. al. had a simplistic, facile conception of pain and pleasure that didn't take account of the way the boundaries between the two are always in flux, and that in some cases, it wasn't for anyone else to judge what constituted unbearable suffering for another.
When people try to benefit someone in distress, the intellectual frivolity with which those moved by pity assume the role of fate is for the most part outrageous; one simply knows nothing of the whole inner sequence and intricacies that are distress for me or for you. The whole economy of my soul and the balance effected by "distress", the way new springs and needs break open, the way in which old wounds are healing, the way whole periods of the past are shed – all such things that may be involved in distress are of no concern to our dear pitying friends; they wish to help and have no thought of the personal necessity of distress, although terrors, deprivations, impoverishments, midnights, adventures, risks and blunders are as necessary for me and for you as are their opposites. It never occurs to them that, to put it mystically, the path to one's own heaven always leads through the voluptuousness of one's own hell. No, the "religion of pity" (or "the heart") commands them to help, and they believe that they have helped most when they have helped most quickly.If you, who adhere to this religion, have the same attitude towards yourself that you have toward your fellow men; if you refuse to let your own suffering lie upon you for even an hour and if you constantly try to prevent and forestall all possible distress ahead of time; if you experience suffering and displeasure as evil, hateful, worthy of annihilation, and as a defect of existence, then it is clear that besides your religion of pity you also harbor another religion in your heart that is perhaps the mother of the religion of pity: the religion of comfortableness. How little you know of human happiness, you comfortable and benevolent people, for happiness and unhappiness are sisters and even twins that either grow up together, or, in your case, remain small together.
Of course, a lot of people are accepting of the idea that pleasure and pain are two sides of the same coin, and that it's impossible to have one without the other. After all, Nietzsche's own maxim that "what doesn't kill you makes you stronger" has actually become a worn-out cliché in our time. A different twist that I see a lot, one that I think needs addressing, is the rationalization technique in which one tries to retroactively define negative experiences as positive ones, blessings in disguise. Sometimes you hear people say that some horrible experience was "actually the best thing that ever happened to me!" Well, no. If we're going to turn negatives into positives, the negatives have to be allowed to remain negatives. That's where their power derives from, so to speak. Claiming that everything is good in one form or another is like saying that "everything is up". It makes the term meaningless. Plus, it's just semantics. You can't define or rationalize away the immediate experience of pain or fear, something no one enjoys at the time. You just have to accept the inevitability of it and cultivate techniques for getting through it without coming to pieces; the benefits come later. There's no contradiction in being glad for a useful lesson derived the hard way while also admitting that you would never willingly choose to experience such a thing again.