Predicting the future is notoriously difficult. Towards the end of the 19th century, the famous physicist William Thomson, more commonly known as Lord Kelvin, proclaimed the end of physics. Despite the silliness of declaring a field moribund, particularly one that had been subject to so many important developments not so long before Thomson's ill-fated pronouncement, you can't really fault the poor devil for not foreseeing quantum mechanics and relativity and the revolutionary impact they would have. Seriously, how could anyone, even someone as smart as Lord Kelvin, have predicted quantum mechanics?
So I'm not going to even try. I'll stick to a safer (and more prosaic) prediction that has already begun its realization. Increases in computing power, in part through shared computational resources, are likely to transform the nature of science and further revolutionize the spread of information. Individual computing power might increase according to Moore's Law but a more discrete jump in computational power should also result from clever uses of computers in concert.
Already we have seen SETI allow for a large-scale search for extraterrestrial signals that would not be possible with any individual computer. Protein folding is currently being studied through a distributed computational effort.
Currently CERN is developing "grid computing" to allow the increase in computational power that will be required to analyze the enormous amount of Large Hadron Collider (LHC) data. Though the grid system would be hard pressed to have the transformative power the World Wide Web (also developed at CERN), the jump in computational power that can be possible with processors coordinated the way that data currently is can have enormous transformational consequences.
Modern science has two different streams that face very different challenges. Physicists and biologists today, for example, ask very different sorts of questions and use somewhat different methods. Traditionally scientists have searched for the smallest and most basic components from which the behavior of large complex systems can be derived. This mode has been extremely successful in understanding and interpreting the physical world.
For example, it has also helped us understand the operation of the human body. I am betting this reductionist approach will continue to work for some fields of science such as particle physics.
However, understanding some of the complex systems that modern science now studies is unlikely to be so "simple". Although the LHC's search for more fundamental building blocks is likely to be rewarded with deeper understanding of the substructure of matter, it is not obvious that the most basic structure of biological systems will be understood with as straightforward a reductionist approach.
Very likely individual elements will work in conjunction with their environment or in collaboration with other system elements to produce emergent effects. Already we have learned that the genetic code is not sufficient to predict behavior, but gene's environments that determine which genes are triggered also play a big role. Very likely understanding the brain will require understanding coordinated dynamics as much as any individual element. Many diseases too are unlikely to be completely cured until the complicated dynamics among different elements is fully processed.
How can massive computing power affect such science? It will clearly not replace experiments or the need to identify individual fundamental elements. But it will make us better able to understand systems and how elements work in conjunction. Massive simulations "experiments" will help determine how feedback loops work and how any individual element works in concert with the system as a whole. Such "experiments" will also help determine when current data is insufficient in that systems are more sensitive to individual elements than anticipated. Computation alone will not solve problems — the full creativity of scientific minds will still be needed — but computational advances will allow researchers to explore hypotheses efficiently.
At a broader level (although one that will affect science too) coordinated and expanded computational power will also allow a greatly expanded use of the huge amounts of underutilized information that is currently available. Searching is likely to become a more refined process where one can ask for particular types of data more finely honed to one's needs. Imagine how much faster and easier "googling" could be in a world where you "feel lucky" every time (or at least significantly more often).
The advance I am suggesting isn't a quantum leap. It's not even a revolution since it's simply an adiabatic evolution of advances that are currently occurring. But when one asks about science in twenty years, coordinated computation is likely to be one of the contributing factors that will change many things — though not necessarily everything.