LoK has a more serious tone, but I don't believe that makes it darker, even when it tries to be. A:TLA really hammered home the realities of war and loss, and nothing LoK's done has reached that emotional depth, honestly.
Legend of Korra tries to be mature, but fails horrible at it,
LoK thinks it can be mature by dealing with "mature" topics. Politics! War profiteering! Civil unrest! The show believes it can be grown up by just wearing dad's pants, but it rarely does the legwork to really complement the themes it's trying to push.
AtLA was so successful because it built this fantastic world with a really interesting and understandable backdrop that drove the main plot at all times. Then the show built well realized characters within this context. And it had these characters play off one another effectively so that AtLA's themes of understanding, empathy, friendship and forgiveness could be pushed. They're not "mature" themes on the surface like what LoK wraps itself in, but they're qualities every mature person would appreciate, and are the soul of the entire show. Much care and respect is given to them.
For example, look to how this thread was begging for Korra and her gang to split up so that each character could be apart from each other. Their interactions were that toxic. The plot would struggle because everyone would be stupid and nothing could be built between them. Leaps of logic would happen all the time, and nobody learned. Contrast that to Aang and his friends: even when they would fight, there would be an impetus behind the argument which led to a resolution that makes sense within each respective character.
And the plot was driven forward.
You hardly see that in the Legend of Korra. Mako in particular was a character leech for far too long before he became merely tolerable. Satch has a really good post detailing why he was so bad - if someone would be so kind to link it.