Historically, governments sought to spur innovation and the development of solutions to important social problems by offering prizes (including large sums of money) to the first person to solve the problem. As noted in this Daniel Drezner post (and in more detail here and here), this is an effectgive way to spur innovation, but not nearly so effective at meeting political demands. Subsidies and grants are far more popular for politicians, but it's not so clear they produce the same social benefits. Among other things, they encourage the politicization of science, require expenditures irrespective of whether a problem is solved, exclude potential sources of innovation, and fail to take advantage of dispersed knowledge. The problem, however, is that government subsidies and grants are easier to administer and in the interest of the political class.
Forget Subsidies, Try Prizes: