I don't think Hitchens is right to call the Whites fascists, as he essentially does here. And while the Whites could be incredibly brutal, and often were, I wouldn't say the difference between White and Red brutality was the scale, but the motivation and direction of it.

The White regimes were quite bad at governing in practice, but in theory they weren't worse than their rivals. All the major White regimes claimed to be fighting in the name of Russian democracy and the Constituent Assembly, dissolved by the Bolsheviks. But in practice, the White governments often gave high-ranking positions to bullies and lackies who extorted and terrorized their charges. White official policy in any case was at its worst on labor issues. In Kolchak's Siberia strike agitators were shot on sight. In Siberia large anti-White partisan movements rose up in 1919, which shows the sheer unpopularity of the White government there.

In White South Russia Jews suffered especially at the hands of White soldiers, notably Cossacks, who murdered, raped, and stole from Jews in occupied towns. This was not official policy, however, and General Denikin, the leader of the White Volunteer Army, did take steps to curtail it.

The Reds brutalized their own populations too, executing suspected political enemies and confiscating food, leaving villages to starve. However, Lenin's Red Terror was better directed and controlling people and supplies through fear. There were also revolts against Bolshevik rule, like in Tambov in 1920 where tens of thousands of peasants fought against the Red Army, which responded with mass executions and poison gas. The difference I think was that the Red Terror was organized and directed at controlling the population, whereas the White Terror happened because of governmental neglect and the role of individual soldiers.

As a final example, both the Reds and Whites were often horrible to their own soldiers and to POWs, but again the Reds enacted their brutality with more discretion and direction. The Reds, infamously, used blocking detachments to machine-gun their own troops if they fled in battle, but also tried to motivate them politically. The Whites, on the other hand, were often needlessly horrid to their own men. A delegation from a regiment in Kolchak's Army, for example, protested to their commander that they had not been feed for days or issued boots and overcoats. Their commander had 20 executed on the spot and flogged the rest. Unsurprisingly, when the regiment arrived at the front, it duly deserted en masse to the Reds. On that note, the Reds treated rank-and-file White prisoners well, with Trotsky ordering that the by treated as friends, and not summarily exectued, to try and induce as many as possible to join the Red Army. (They did, however, usually torture and murder captured White officers.) The Whites on the other hand usually executed captured Red units en masse, sometimes simply by stripping them of their clothes and leaving them to freeze to death.

TL;DR: No, both sides were nasty, but they were a little different in their nastiness.