Others have already pointed out the obvious magnitude-of-funding issues.
Another issue though is that all of humanity benefits from scientific advances. If government funding were to reduce and be replaced by fund raising drives, then (in the simplest case) those who don't contribute would be getting all the benefits (alternatives to fossil fuels, medical advances, etc) but with none of the upfront cost. Of course, we already have some fund raising for breast cancer/prostate cancer/MS/other specific disease but I would imagine this makes up a fairly small portion of their research budgets (and in some cases genuinely represents an investment in their personal future).
The obvious way around this is through a Kickstarter style reward system, where people who contribute get some specific rewards. But what would you offer? You get a share of the profits? (Well, now you're actually a corporation.) You get early access to the treatment? (That's not going to fly politically.) You get your name on the side of the particle accelerator? (That might work.)
Obviously, people are welcome to do whatever they want with their money, but I think government funding of science for the common good is the fairest scenario, and what we should be encouraging.