Yes, we can (probably) do more science and for cheaper with robots. Yes, we can (probably) develop the same technologies we would other wise by focusing just on automated exploration but there is one very important thing only human exploration can do:
Inspire generations of future scientists
Never underestimate the power of seeing someone personally going where no one has ever gone before or doing something that has never been done before. Thousands of people cross the straight between England and France everyday, but when someone swims across it makes the news. Thousands of people skydive everyday, but when Felix Baumgartner jumped from higher and faster than anybody before it made the news. How many of the older generation of scientists and engineers chose (at least initially) that carrer path because they grew up watching Neil Armstrong bounce around on the Moon? Or watching the now defunct Space Shuttle take people to the edge of space and dreamt of doing the same? A big project similar to the space race could make science and engineering sexy again, and would have one other advantage that only giant projects with clearly defined goals have:
It focuses research in one direction, speeding up technological evolution
The reason why a bunch of nerds in the middle of the New Mexico desert were able to build an atom bomb in practically record time was because everyone had a clear idea of what the goal was and it was immediately clear when progress was being made towards it and when it was achieved. The same thing applies to the Space Race. When Kennedy first said "we choose to go the moon in this decade" it was far from clear that it would even be possible. But it gave everyone a clear and inspiring goal to strive for. "We choose to send a bunch of circuits to the moon" wouldn't have nearly the same effect. This is the reason we now talk of "moonshot projects" (and google has a whole division for them). They are almost impossible but the goal is sufficiently clear and inspiring to make people work hard until they are accomplished.
Imagine what would be possible if Obama were to say "we choose to go to Mars before the end of the decade" and actually provided NASA and private contractors with the funds to do it? What technologies wouldn't be developed and commercialized that we haven't even dreamt of?