The simple answer, of course, is that yes there is a strength difference. How much of a strength difference, and whether it is significant, depends on how you design your structure and how the load is applied to it.
To be completely technical, strength refers to either the Ultimate Tensile Strength (where the material breaks) or Yield Strength (where the material deforms and does not return to it's initial shape). When you bend a paperclip into a new shape you have exceeded the yield strength (also known as yield stress) but not the ultimate tensile strength (or UTS).
What you are most likely intersted in for your robot, however is not yield stress, or UTS, but rather stiffness... how much the structure will flex under a given load. Rather than go into great detail, I'll suggest you look at the manual for West Point Bridge Designer (a great program and a great contest) at
http://bridgecontest.usma.edu/
Suffice it to say, however, that the 1" tubing will be significantly more stiff than the 3/4". In most cases, however, it is not likely to be the tubing that fails... you are far more likely to get failure at a poorly designed joint or corner. Don't be afraid to ask the person doing your welding if they think the joints are strong enough. Someone with the skill to make a good weld in aluminum tubing has seen enough joints to know where a little extra gusseting is needed.
For the record, our robot will likely use mostly 3/4" by .062 tubing in the superstructure on top of a KOP chassis.
Good luck, and DO check out the West Point Bridge Designer site. Steve Ressler at West Point has done a fabulous job of explaining structures and materials in a way that is high-school student friendly.
Jason