Originally Answered: Unmanned space exploration?
First of all, I would like to say that I am a big believer in manned space exploration. I feel that robots can get lots of great science results, but there must still be progress for humanity in its ability to become a starfaring species. A journey of a thousand miles begins with one step, and we're on step 14. We should definitely continue on this front.
However, for scientific gain, unmanned missions are much more efficient. Firstly, robots can survive in much more extreme conditions than humans. The radiation environment around Jupiter would make manned exploration of the Jovian system nearly impossible for any duration.
Robots only need electricity to "eat" and don't need breathable air or water to keep them going. This cuts down the amount of weight needed to launch, which keeps costs down. Look at the launch vehicles used to send our fastest space craft yet (New Horizons en route to Pluto) and the Apollo missions. The Atlas/Centaur launch vehicle is much smaller and requires much less fuel to go directly to Pluto than the Saturn V did to go only to the Moon. This is because all of the water, food, life support systems, equipment, etc necessary to keep 3 men alive for only 2 weeks required all that extra energy just to get it away from the Earth. Robots don't need all of that.
Another major advantage with robots is that they can have a much longer mission duration than manned missions could. Take the Mars Exploration Rovers, Spirit and Opportunity, for instance. Their stated mission duration was 90 sols (about 92 Earth days). They have been working and returning valuable scientific data for over 2 years and they're still going. This does not include the cruise time getting to Mars.
Robotic missions also present less risk in public support because they're cheaper and failures don't cause nearly the public outcry or spectacle that losing human life in space would cause. Imagine if Neil Armstrong had crashed the LEM during his historic descent into Mare Tranquilitas. Everyone around the world would have witnessed it live on TV, and most people would have questioned wheter or not we should be there, etc. When Mars Polar Explorer crashed because of a mixup in use of Imperial vs. Metric units, it was a black eye to NASA, but nobody died - only a machine.
Since these missions can be performed with less cost and smaller rockets and less materials, more of them can be flown. NASA's budget is already miniscule on the scale of our government's budget, and this would not allow us to be doing very much out there in the outer solar system, maybe one mission per decade if even that. As it stands, we have several missions active on Mars, as well as probes en route to or active at Mercury, Saturn, and Pluto. Even Voyager is still alive and kicking at over 100 times the distance from the Sun as Earth, telling us about the nature of the space environment at the edge of our Solar System. We could not have kept humans alive in space for their amazing 30 year trip.
I think that the major goals of manned space travel are similar but definitely separate from those of unmanned space travel, and that both should be continued, and with more effort than we are putting into them today. While manned space travel is about learning to survive in space and eventually other planetary environments, for purely scientific missions, robotic travel is the way to go.