Gasoline is made up of the petroleum fraction that boils below 200 degrees centigrade (390 F). Aviation gasoline has a smaller boiling range (38-170 C, 100-340 F), leaving out the lowest boiling components that are in auto gas, largely because of extreme volatilities they would have at the altitudes involved in flying.
The two tests used to determine "research" and "motor" octane differ in the load on the test engine (more load for the motor test). Both octane and cetane tests are described by, and conducted according to specifications of, the ASTM (American Society for Testing Materials). The standard test compound is "iso-octane" as oil men call it. Chemically it is not iso-octane which would be 2-methylheptane, but rather 2,2,4-trimethylpentane, a highly branched eight-carbon hydrocarbon.
Gasoline engines knock less on branched hydrocarbons, although the straight distillate of raw petroleum tends to contain mostly straight-chain hydrocarbons in this low-molecular-weight range. Cracking and catalytic reforming processes are used to increase the percentage of branched hydrocarbons to improve octane ratings.
Aviation gasoline usually has no olefins (alkenes) because they tend to form gums and have poor antiknock characteristics. Aromatics, such as benzene and toluene, have good octane ratings under load (rich conditions) but act more like olefins under lean cruising. Toluene has research/motor octane ratings of 120.1/103.5; benzene has 114.8 motor octane, compared to "isooctane" which is set arbitrarily at 100 on both scales.
In 1922, tetraethyl lead was found to improve the anti-knock characteristics of gasoline. This became more important in the 1930s because the increased demand for gas led to use of cracking processes that produced more gasoline from crude oil, but of lower octane ratings. Standards for octane ratings over 100 are made from "isooctane" with tetraethyl lead added (1% = 108.6; 2% = 112.8; 3% = 115.5, etc.).
Crude oil has more of the branched, cyclic, and aromatic hydrocarbons in the higher molecular weight range where Diesel fuels are obtained. Diesel fuel, and fuel oil, have a boiling range of about 175-345 C (350-650 F). The standard for Diesel fuel ratings is "cetane" or n-hexadecane. This is a straight-chain, 16 carbon hydrocarbon with a short-delay period during ignition, and its rating is set at 100. Heptamethylnonane is a highly branched 16 carbon hydrocarbon with a long-delay ignition, and cetane rating set at 15. Diesel fuels largely contain molecules having 10-20 carbons, whereas gasoline components have mostly 12 or fewer carbons. Diesel fuel power in terms of heat content is increased by saturated hydrocarbons, but these are prone to form waxes at low temperatures. Ignition performance is improved by straight-chain hydrocarbons, such as cetane.

As mentioned above, crude oil is just the "opposite" of what we want--it has a lot of straight chain small molecules where we want branching, and it has a lot of branched, cyclic, and aromatic (highly unsaturated) heavy molecules, where we would prefer straight-chain saturated molecules. One legitimate reason for Diesel fuel price increases is the cost of removing sulfur to meet EPA requirements.
The bottom line is that the best Diesel fuel would have a lot of "waxes" or saturated, straight-chain molecules, up to the limit of the cloud point and pour point allowed by ambient conditions. The other "stuff" helps with viscosity, pouring, lubricity, etc. but is largely there because that is what is available. It should be apparent that a poor Diesel fuel would be made up of small molecules with a lot of branching and unsaturation--that is, a pretty good gasoline!

To use our normal frame of reference, we know that gasoline ignites very easily, and is very volatile. Diesel fuel is much less volatile--it stays on you when you spill it during fueling the truck, even after you try to wipe it off. Diesel fuel also ignites much less easily. So, if you put some of the above "pretty good gasoline" in your Diesel, it would ignite so explosively that the heads would pop off the engine, etc. That would be "powerful" but in the explosive sense. It would have less heat content (the useful kind of power), because of the smaller, unsaturated (aromatic, etc.) molecules, so it would decrease fuel mileage, if the engine could stay together. Now you see why #1 Diesel and winterized Diesel fuel decrease your fuel mileage. To improve the cloud and pour points, lower the viscosity, and increase volatility to compensate for low ambient temperatures, smaller molecules, and ones that tend to stay liquid at lower temperatures (branching and aromaticity help here) are used in the fuel. They do the needed job, but have less heat content (often expressed in BTUs or British Thermal Units). Basically, the characteristics that make more power are more carbons and more hydrogens per gallon, and saturated molecules have more hydrogens.