Gasoline is a light molecule, so it is volatile...meaning it has a high vapor pressure...the amount of pressure in a container due to the vaporized material relative to other materials (the atmosphere- N2, O2, CO2...). That's why it smells so strong and evaporates so quickly...and goes BOOM so easily. Diesel, having a low vapor pressure, does not normally ignite when a match is thrown on it. Care to try that with gasoline?

A container at constant total pressure (your gas tank) can hold a more or less fixed amount of stuff. If instead that volume is filled with liquid (or solid), then not much gaseous/vaporized stuff can fit. Your fuel remains in the liquid state, useable for driving.

If its mostly not filled with liquid, then there's a lot of space to fit a lot more vaporized gasoline...not useable for driving but great for adding to envorinmental problems. That's why vapor recovery is important.

Now, as far as getting better fuel efficiency, that's not what we care about. A full tank weighs more...higher pressure at the tank bottom where the pump takes suction (think about your ears at the bottom of a pool). The fuel pump adds pressure to that, and eventually its squeezed into your cylinder where it vaporizes as it is released into the low-pressure of the cylinder chamber and ignites.

So a full tank can deliver slightly higher fuel pressure to the cylinder, so when the gas gets to your cylinder, it evaporates more easily. Less energy required to make it go boom means more energy available to go roundy-roundy. There's more to it than that, but there's part of the just of it. Same is true for any pumping system- the more "head" you can deliver without having to acutally pump, the more efficient the system is.

A full tank also compresses the fuel ever so slightly, meaning more gasoline 'fits' in a gallon at the bottom of the tank when full than empty...we're talking a very small amount, though. Hydraulics guys, don't kill me here- gasoline is not a perfectly incompressible fluid. On the micro scale it does pack in there more tightly.

I'm not sure how much difference it makes, but its for real. Methinks you gotta have some very precise instrumentation to see the effect on the scale of a single car engine.

DISCLAIMER: This is the THEORY. I don't go around measuring fuel manifold pressures on full/empty tanks, so I cannot back it up for cars. Ships, yes. Cars, no. As I said, the difference would probably be hard if not impossible to see; I tend to give credence to the nonlinear gas gage thread, making people think they get better mileage.