The first advantage is straightforward: a feed-in tariff sets the price for energy from a given type of renewable source so that regardless of changes in market prices of competing options, those who invest in that particular technology have a guaranteed price for their product for many years into the future. That removes a lot of the risk and uncertainty inherent in energy investments, and greatly accelerates the level of deployment of these technologies. This of course requires the feed-in price to be high enough that investors will actually make a profit if their basic costs and projections pan out; over time the feed-in price can be reduced as technology improves.
The second advantage is that this doesn't (necessarily) involve any additional use of government funds - the money comes from the utilities, and in most cases is just passed through to electricity consumers. This is forcing consumers to support renewable energy, but at least it is in a fair manner that applies across all who receive power from a given utility, and those who use less pay less. In any case, the "bang for the buck" at the government level of such a program is enormous, compared to other alternatives that provide more direct subsidies (like the PTC).
Experience with feed-in tariffs in Germany and Spain has been phenomenal - by 2006 roughly 10% of Germany's electric production was from new wind and solar installations; England is now embarking on similar regulations. Taking this route in the US would greatly accelerate the already rapid growth of wind and solar in the world's energy supply. This both has the immediate impact of reducing CO2 emissions from displacement of fossil-fuel plants, and also brings us necessary experience in large-scale deployment of these technologies. Once manufacturing capacity and demand is at a level that can quickly satisfy a significant percentage of US electric supply, we will have to make additional choices of the most cost-effective paths to complete replacement of fossil energy. But up to that point, feed-in tariffs are clearly the best way to greatly expand renewable supply.