Utilitarianism suffers from two weaknesses:
The first is derived from utilitarianism's greatest strength: It aims directly toward maximizing goodness (or Quality, or whatever you'd like to call it). This is, I think, the secret aim of all other philosophies. Another philosophy might say that "Doing X is the best thing you can do." It makes X the ostensible goal. But it justifies that goal by stating that it is the "best;" i.e., the most good. This unquestioningly implies that whatever action is the most good is what one should do. The proponents of that philosophy may go to great lengths to prove that X is the best action, never noticing that the goal of X, and thus their ultimate goal, has been goodness the whole time.
But in driving right to the heart of the matter, utilitarianism runs into the same difficulty already known to Buddhism (which also acknowledges goodness directly): The Good cannot be defined. This leaves utilitarianism in a seemingly awkward position, compared to most western philosophies. Its primary goal appears nebulous, while the others are fixed upon narrow paths whose goals, while definitively incapable of being as worthy as goodness itself, are frequently better defined and thus more comprehensible.
Many utilitarians circumvent this vagueness by using happiness as a measure of goodness. (That is, the quality of an action can be measured by the happiness it creates or preserves.) This is a generally useful schema, though it does leave them open to attacks (which are weaker than they seem1) based on the imperfect correlation between goodness and happiness. Rather than bow to this imperfection, John Stuart Mill expanded his definition of "happiness" to include other significant results of good actions, such as intellectual satisfaction, pleasure, and freedom from pain. But he got caught up in the problem of how to decide what is best when there is disagreement, because he still hadn't defined "good". So he fell back upon trust in the opinion of the majority, a method that has since been discredited by U.S. elections. No other easy schema could have sufficed: Any that could, would have provided a simple definition for "good" (which we already know to be an NP-hard problem2).
The second weakness of utilitarianism is its lack of guidance. It provides a goal, but no means to achieve that goal. Again, this makes sense, given that goodness itself varies with people's situations (internal and external). Even aside from that, no one lifestyle can maximize goodness for everybody, given that people have differing resources and abilities. Yet, guidance is appealing in a philosophy, and occasionally even useful. It is certainly possible to compile sound philosophical advice, and even to tailor it to the needs of the individual. Such endeavors are utilitarian simply by being worthy of doing. Yet the advisory content of such compilations seems to lay outside of the scope of utilitarianism. (On the positive side, this lack of completeness places it ahead of those philosophies and religions whose advice is useless or harmful. An independent utilitarian is more likely to do good than an organization is.)
1. For example, there's the "happiness pill" argument. If we had a cheap pill with no harmful side effects, which caused blissful happiness, would a society of people who spent their entire lives experiencing nothing but this drug-induced happiness be the best thing possible? The questioner knows perfectly well that it is not (even if they are unsure as to why), and hopes to lure the utilitarian into admitting that a philosophy of happiness could lead to a result far removed from goodness. The simplest response is that such a society is not the best thing possible, on the absurdly simple grounds that it is not "possible" at all. For one thing, such a society is unsustainable; its populace would have its basic needs unmet. (Every counter to this objection that I have heard involves robots.) For another, such constantly euphoric people would never develop an emotional capacity for happiness; they would never move beyond the capacity for raw pleasure.
2. This is a metaphor, referring to a class of problems that are all difficult (time-consuming) to solve. If a fast solution for one of them is ever found, it will bring about fast solutions for all of them. To show that a problem is in this class implies that a fast solution for it has never been found, and cannot be found any more easily than can a fast solution for any of the others. The problem of finding a simple schema to always determine what is best, and the problem of defining "good," are similarly intertwined. A limit to our ability to define "good" limits how satisfactory any such schema can be.