Perhaps I'm just not seeing the distinction, but can't they be the same thing? You can use an "idea" to solve a "problem". An idea (whether self generated, or off a list from pg's head) could be rooted in a real-world problem.
idea 1 = reel to reel tape
idea 2 = self contained cassettes
idea 3 = hard disks
idea 4 = floppy disks
idea 5 = flash based drives
If you concentrate on the problem (we build the most reliable/fastest/cheapest devices for persistent storage) instead of the idea (we build the most reliable/fastest/cheapest tape drives) then you're less likely to be screwed when the market or technology changes
The benefit to targeting a "problem" is that you have a preexisting need. An "idea" with no existing "problem" may require several years of extra lead-in where you are struggling to get people to understand what you do and why it should matter to them. It can work, but the ratios are less favorable for a VC-style investment which looks for fast turnaround with minimal risk and orders-of-magnitude payoff.
But the key is to always have a set of problems floating in your head. Having an idea, and seeing what problems that idea can solve is not as effective, as you often end up trying to force a square into a hole.