The unity of Red forces as opposed to the disunity of White forces was indeed an important factor in the White defeat. This manifest itself in both physical as well as ideological unity. Specifically, the Red Army controlled a central, unified territory. This meant they could communicate, move troops etc. in an organized fashion. White troops, on the other hand, were located in separate pockets on different sides of the Russian Empire (Southern Russia, Siberia, etc). In fact, it makes sense to speak of the White Armies, rather than a single unified White force. These different groups, led by different Generals, had trouble communicating, and therefore coordinating attacks.

Furthermore, the lack of ideological unity -- as u/Iisdabest889 mentions, 'White' was a catchall for everyone opposed to the Reds, while the Red Army was fighting FOR a specific, communist cause-- made it difficult to unite the different White forces. In fact, Russian historian Oleg Budnitskii has argued that the one idea that linked different White forces was antisemitism. As a result, even when it hurt the discipline of the fighting forces or alienated locals, White Army forces often prioritized attacking Jews and looting their homes or businesses over better strategic decisions, because it was a morale boost or unifying exercize. (See Russian Jews Between the Reds and the Whites).

However, it is important to point out that many historians today argue that the Red Army did not so much win the Civil War as the White Army lost the war. In 1921 a famine broke out, which caused many already exhausted people to refuse to fight. The entire country was unable to continue the war, and at that point the Reds remained in control of more, and more significant, territory (including the capitol). The White forces fell apart, and the Red Army held out slightly longer, so ultimately won.

Slightly divided is a real understatement. The Whites were anyone who wasn't a Bolshevik, effectively. This meant agrarian revolutionaries and conservative monarchists were to fight on the same side against a common enemy, despite their total ideological opposition to each other. Even though they had combined their forces, they attempted to undermine each other at every turn (IE the liberal commanders may not inform the socialists or monarchists that there was a planned assault on one of their zones). No such divisions existed in the Red forces, who were united under one group.