Will I notice any significant difference between 700x32c tires and 700x35c tires in terms of speed/ease? Thanks!
A couple of decades or so ago, some well meaning engineers did some rolling resistance tests on tires and found that higher pressures led to lower rolling resistance. Their data was undeniable. Unfortunately, their tests were flawed and so their conclusions were wrong. They tested on a smooth surface. I don't know about where you live but a nice smooth road is hard to come by around here. Their conclusions hold up nicely on a nice smooth velodrome. Out on the roads, it's not so simple. More recent testing has shown that the rougher the surface, the lower the pressure you want. High pressure tires apparently fight the little bumps in the road harder than lower pressure tires and end up having more rolling resistance. Narrow tires need higher pressures than fatter tires in order to avoid bottoming out on bumps which will cause pinch flats. Fatter tires also require more material so they are necessarily heavier which could be an issue for climbing. The difference between 32c and 35c is going to be small in any case.
A 3mm difference is noticeable only when there is a large variation in pressure. Assuming since those are 700's that you have a hybrid/commuter and you ride on the road, if the 35 is say an 80psi tire and the 32 is a 110 psi tire, you will notice a difference. The 32 will roll easier. If you do have a hybrid bike I suggest an even larger difference going to 700 X 28 Continental Gatorskin All Season. You will think you bought a new bike as it will pedal much easier at speed than the 35's. The narrower tire is suggested only for paved road usage. It will not handle loose gravel as well as a wide tire.
There is very little difference between these sizes. The wider tire will be a bit heavier, but rolling resistance using the recommended pressure will be about the same, assuming that both tires are the same brand and model. Contrary to what others are telling you, higher pressure may not reduce rolling resistance. Tires of this width run best at between 65 and 85psi, depending on rider weight. Inflating tire too hard on anything but a glass smooth surface will result in increased rolling resistance and a very harsh ride. However, different tires do roll differently. Heavy flat resistant tires will have a harsher ride and more rolling resistance. Lighter weight tires designed for more performance will have a more supple ride and less rolling resistance
