-- 1 Corinthians 8:2
"The only real wisdom is knowing you know nothing."
"He who speaks, does not know. He who knows, does not speak."
-- Tao Te Ching
There's a Zen story about a guy who became enlightened while watching smoke rise from a candle. A scientist might look at a burning candle and think about entropy, heats of combustion, activation energies, turbulent flow, emission spectra, and so on, going into increasingly elaborate and detailed thoughts. A Buddhist might look at a burning candle and think, "everything changes," and then stop there, focusing on the experience of change rather than all the categories of it.
The scientist's thoughts are more useful on a technical level--you need these theories to build certain devices. The Buddhist's thought is more useful on a personal level--"everything changes" is enough to inform the way you act and treat people, what you value and believe. I would say the Buddhist's thought is more valuable overall, because accepting that everything changes, when you really digest it, will determine whether and how you will make use of scientific knowledge.
It doesn't necessarily work the other way, meaning that science is not so personally instructive. I don't think it's because scientific knowledge is incomplete--after all, "everything changes" leaves a lot more unsaid--but because the discipline of science is reluctant to emphasize the impossibility of complete knowledge. There is an implicit promise in scientific study that an answer is just around the corner. Wishful thinking, perhaps. In Buddhism, Zen in particular, there seems to be a central emphasis on the limits of knowledge, and constant admonition against seeking answers in faraway times or places, where mental projections dominate actual perceptions.
Just as important, the two views differ greatly in their intention. The scientist doesn't see a need to stop the craving for knowledge. They just keep on searching like there's a way to escape the unknown. The Buddhist might not be able to stop the thinking, but they can hopefully live with the feeling of uncertainty when it happens. It's not that it's more satisfying or intellectually stimulating to not seek; it's that not knowing is an essential part of life that we have to face whether we want to or not. This is not just part of Buddhist practice; it is a common theme in many religious traditions.
Reality is fully expressed in each moment. If things are unknown and mysterious now, then they're probably like that for a very deep reason; to truly understand how to live in reality, one has to understand how to live with uncertainty. It took western science a long time to accept the fundamental uncertainty inherent in empirical knowledge, but people seem to have realized it centuries ago just by noticing that everything changes.
I wouldn't say that a religion or spiritual tradition such as Buddhism should replace the scientific method. Science is obviously a good way to solve everyday problems in technology, industry, and medicine, but those specific applications are of limited use: science can only make our lives more efficient, more stable, and more comfortable, temporarily. No amount of scientific knowledge can replace the simple experience of reflecting on change and coming to terms with the reality that everything that we know, including our own bodies and minds, will gradually change and go away.
If we use science as a crutch, to distract ourselves from this reality and turn our attention to concepts, then we're not really using it for a good purpose. We might actually abuse the insight that comes from science if we're in denial, if we fail to understand what it means to be fragile, imperfect, and transient. The worst thing we can do with science is use it to make ourselves feel powerful, real, enduring, important. That's how people create problems.
But the great thing about reality is that it tends to teach its lessons one way or another. If a person isn't content to accept the mystery of life at the first opportunity, reality will eventually force them to do it. This can be seen in science. Just look at some of the debunked theories that tried to explain away the mystery of reality with simple and tidy concepts: steady state theory, luminiferous ether, caloric, celestial spheres, the Bohr atom, vital essence theory, and Newtonian gravity, to name a few big ones. Compare these to some of the theories that replaced them: relativity, quantum mechanics, and the kinetic model of temperature. These theories explain everyday phenomena in terms of complex relationships that are only said to exist in terms of one another, and which ultimately stem from mysterious events which can fundamentally not be observed.
At the frontier of scientific knowledge, there is a convergence with the Buddha's teaching that all things lack an intrinsic nature, but this insight has yet to become part of mainstream culture or education, in spite of the importance of science in both. Nevertheless, it's within our power as individuals to make use of this knowledge: in our own lives, we can choose when to indulge in curiosity, and when to live directly with the mystery of human existence.