I personally went from not understanding why demons and devils were different things when I first started playing and running D&D to the Blood War and the stark difference between devils and demons being one of my absolute favorite aspects of D&D lore.
The way I run them, devils are far less likely to fight mortals than demons are. Devils don't want to kill you; they want you on their side and to believe that their way of thinking is how all existence should be based upon (and they also want the credit to get themselves closer to promotion). Battles against enemies other than demons is a fail state; it's a failure to recruit more mortals to the cause of the Hells. Better to prove the necessity of what the devils are doing to at least start mortals on the path or thinking that the fiends are actually just misunderstood.
Devils also do the dirty work of fighting the hordes of the Abyss in the Blood War so that the celestials don't have to. If people weren't selling their souls and becoming devils celestials would have to expend all their efforts fighting demons and risk being corrupted themselves. As a result of this, devils are also less likely to be impulsively cruel; every devil in the Hells is ultimately the property of Asmodeus, and it would be a transgression to waste his finite resources. Cruelty is built into the system of the Hells itself rather than something individuals do to each other.
In short, Chaotic Evil is senseless evil, and Lawful Evil is supposedly "necessary" evil. The devils are the primary force preventing the multiverse from the onslaught of the Abyss, and even the celestials know it. If there's another way to handle the situation no one has figured it out. Asmodeus may proudly consider himself the multiverse's greatest hero for performing this thankless task, self-assured that the vitriol directed at him comes from ignorant fools who would perish under the demonic onslaught if he decided to no longer wage the Blood War.
On a less cosmic scale, Lawful Evil could be the alignment of people who believe that by maintaining order and protecting against destabilizing influences they are serving the "greater good". They are the serious men making hard choices about who lives and dies, whose freedoms to suppress, who is an unnecessary burden on society, etc.