All other things being equal, wider tires have similar or less rolling resistance caused by flex in the casing.
All other things being equal, wider tires are a bit heavier than the same tire in a narrower width.
These considerations are generally true when looking at the same model tire in different widths. However, if comparing two tires that are not the same make and model in different widths, often wider tires are the heavier and thicker and stiffer type which roll more slowly, while it is easier to find narrower tires with thin and supple casings which roll better.
Depending on how closely your tire width matches your rim width, a wider tire may have more aerodynamic resistance at high speeds. A big mismatch in rim/tire width (so the tire mounted on the narrow rim has an 'omega' shaped cross section will generally have more aerodynamic resistance than a rim/tire combo that are close to the same width.
A wider tire will almost always be faster on rough or loose surfaces. A wider tire at a lower pressure will glide over surface imperfections, while a higher pressure narrow tire will tend to bounce you and your bike up and down, which is a loss of forward momentum. On soft surfaces a narrow tire will tend to carve a deeper groove while wider tires will be more likely to float on top, which uses less energy.
The great thing about tires is that they are a 'wear item', so you can try a new pair every season or so, or pop new ones on and keep the removed ones as spares or for riding different conditions. I like to take the perspective that riding enough to wear out a set of tires deserves a reward, and I reward myself with a new set of tires!