SLICK VERSUS PATTERNED TREADS
Practitioners of motorized sports recognized the traction benefits of slick tires a long time ago. Drag racing is the sport in which the most obvious benefits of slick tires are demonstrated, and it only takes a few seconds to see that treaded tires just slow the vehicle down.
While that was evident to drag racers early on, it took a generation for grand prix and rally drivers to figure that out, and another generation before motorcycle racers caught the hint. Where I live, in the San Gabriel Mountains, cafe racers test their skill with high-powered motorcycles, which now use slick tires routinely.
I find it interesting that when I get my Continental GP3000 tires they've still got a tread pattern. Fortunately, most road bike tire makers have come 'round to the realization that slicks are in and tread is out.
Bicycle tires (as opposed to car tires) don't need to have a tread pattern to prevent hydroplaning, as a bike tire's round cross section has the ideal shape for the rain. While it is hard to demonstrate this on the road, I'm told that machines that measure traction show that smooth tires corner better on both wet and dry pavement, and that lean angles can be greater with smooth tires than with treaded.
Smooth tread also offers lower rolling resistance, because its rubber does not deform into tread voids. Rubber does not compress, but it is elastic and it deforms. It changes shape, but not volume. For a tire with an internal tread pattern, the rolling resistance is increased because the rubber bulges and deforms into tread voids when pressed against the road. Tread squirm is one term I've heard attached to this phenomenon, and it is mostly absent with smooth tires.
SMALLER- VERSUS LARGER-WIDTH TIRES
I've heard, though I haven't seen, reports of rolling resistance studies Continental performed that included an analysis of tire width. While I've heard about the conclusions of those studies, it's only third-hand, so I can't report them here with any authority.
I've heard, and it seems intuitive to me, that there is not much if any measurable difference in the rolling resistance of a 20mm tire versus a 24mm tire, all other things equal. Thinner tires require more attention, though, in that they're more susceptible to increased rolling resistance if they're not inflated to a sufficient pressure.
I'd be more concerned about the rolling resistance of tires that have some profile other than round. Certain tire makers have tread molds that cause the tire to have a flatter aspect, perhaps in the hope that they'll corner more securely. Whether or not that actually serves the intended purpose, the question is, what does such a profle do to the profile of the contact patch when the bike is going in a straight line?
This is easy enough to test, though it'll require a bit of clean up. Apply a thin layer of wet paint or ink on your garage floor. Roll the tire over it with your weight on the bike, and see what kind of impression it leaves after a full revolution (that won't mean much to you unless you compare the ink profile to that of another tire).
After you've satisfied yourself (or if you're willing to take my word for it) that a 23mm or 24mm tire will roll as efficiently as a 19mm or 20mm tire, you can move to the next two issues, which are aerodynamics and resistance to flats.
As to the former, it depends on the wheel you're using. Remembering that what's aero seems to change every six months according to the various "experts," I distinctly remember a series of conversations between both those who make, and those who wind-tunnel test, aerodynamic bicycle wheels, in which one of the significant parameters was the match between the width of the tire and of the rim upon which it was mounted. In most cases and with most carbon race wheels, 23mm and 24mm tires appear to blend more cleanly than do 20mm tires and thinner.
You're also less likely on both clincher and sew-up tires to flat with a wider tire.
700C VERSUS 650 WHEELS
There has been no bike-related issue that has aroused more debate in the sport of triathlon than wheelsize. Those arguing against 650c always bring up the increase in rolling resistance. The problem with that argument is that I've never seen any evidence to back it up.
The most obvious test, assuming you don't have sophisticated equipment to machine-test the issue, would be a roll-down test of two bikes that are equipped otherwise identically. The problem with this test is that there is one other variable—wind resistance. While this test wouldn't isolate rolling resistance, you could at least determine the aggregate difference (or lack thereof) in the wheel sizes.
Just outside my house is a descent of three-quarters of a mile in one direction, and five miles in the other. Mark Montgomery and I ride from the property several days a week, and we always start off with a roll-down test. I've ridden any of a half-dozen—or perhaps eight or nine—bikes, my own or one of several I've road tested for Slowtwitch or Triathlete Magazine. All of them have had 700c wheels. Monty is always riding a 650c-equipped Cervelo P3. We always pump up the tires to the same pressure.
I am always outrolled, on every bike, and by a significant distance. My consolation was that he weighed more than I—always the most impactful paramater in a roll-down test. Except that last week we both got on the same scale and he was three pounds lighter than I was.
Surely this isn't a scientific study. Maybe his head is more aero, or his rear end has a better trailing edge. Or maybe it's the aerodynamics between this frame and every other that I've been on. I don't know. Monty is certain it's the wheel size. I'm not convinced, but each time he outrolls me I scratch my head and wonder whether it's a parlor trick, or if not, what?
If there is a difference in rolling resistance between the two wheels, it would only be manifested on exceptionally rough roads, in my opinion. Until I see a test that demonstrates the opposite, I'll continue to assert that there is no measurable difference in rolling resistance between the two wheel sizes.