I think I don't say the most important thing: IMO, L&H's comedies are deeper than Chaplin's.
I'll try to explain it in a few words.
Chaplin says things that everyone expects to hear. "Be good", and all that moralizing things. He argues that speech as a way for people to be integrated into a hostile world and transform it. In Easy Street, for example, that policeman transforms all the people in the neighborhood with his good actions.
The Tramp. Chaplin's character, is a marginal one that always ends integrated to the world.
We know that reality is not that simple thing.
L&H try to join the world, to integrate into the system, and simply they can´t. The world is too crazy for them or they're too crazy for the world, but there isn't possibility for them to integrate this world (or being functional to this world, to this system).
Then, although we are integrated in the system, L&H appeal to that part of each one of us that tells us from within ourselves, "do not belong!".
It's a most modern concept that those Chaplin managed.
Everyone can read about it in Freud's Civilization and its Discontents. But it's not necessary to understand this, nor L&H comedies. We all know those aspects of ourselves, like "Tomorrow is my examination, and I am here playing football instead of being home studying...".
I think it is the reason why L&H are in the cover of Beatles' album Sgt. Pepper, and Chaplin is not there.
Chaplin sees a world that he can explain and a world that can be better.
L&H are pessimistic (and Freudian too). They see that the world doesn't work. And can't work ever cause the human being is a totally crazy one. I think, for example, in that film Tit for Tat. It's not a great film, IMO, but, look at this: L&H open a shop and they gradually pay more attention to the fight with his neighbor than to the shop. And people is stealing all their merchandise!!! This idea is more crazy, abstract and complex than all Chaplin did in his life.
Chaplin never could say anything about that thing within ourselves that is disfunctional to the world, cause he was entirely functional. An he won. And he was a millionaire in that way.
In L&H, Ollie's character try many times to live according with social norms and sometimes try to guide Stan to the same way (like parents do with a boy), but it doesn't work. He simply ends following Stan, that incredible adult-child that only does what he can do.
Both of them are two aspects or tendences of ourselves that we can recognize more frequently into us (but it's hard to admit) than that "good like an angel" Tramp that Chaplin shows.
Moreover, the emotional parts of Chaplin's films resort to low blows, like Disney's Bambi or Dumbo. Who's able to not cry (or, at least, to feel emotions) when the subject is so sad as death, hungry or people sleeping outside?
These are easy resources to any writer.
Chaplin was very complete (director, musician, actor, agile and strong like a comedian...) but very overrated about his simplistic vision of reality, like Hitchcock. Their ideas were manichaeist, simplistic, with a reality divided in two types of persons: good people and bad people.
L&H instead, are underrated. There are many more slapstick scenes in Chaplin than in L&H films. Chaplin eating a shoe in The Gold Rush is a great creation, but it isn't very comic. But Stan in Twice Two complicated by the napkin, tie, butter, toast and knife... it is comedy. It's not slapstick, and it's frequent in L&H films. And it's a sharp and deep idea about the world: How many objects that came to help us, come, really, to complicate us? Ask to many mature persons about computers, remote controls, etc.
L&H humor is full of deep observations about the nonsense of today's world.