Avatar: The Last Airbender created its own unique forms of martial arts that The Legend of Korra evolved, with the unique styles of each recognizable to any fans of the franchise. While much of the original series' bending styles drew from traditional martial arts there was a clear influence of mixed martial arts, boxing and other modern fighting styles in the sequel series that tended toward a sleeker and more efficient look. But that does not always mean they're better. In fact, the older bending styles might actually be more effective.
It's only natural for Avatar fans to compare which style of bending is better between the series. The original Avatar brought together many of the franchise's most adept Benders, with characters like Aang, Toph, Katara and Azula using their skills to show off the versatility of their respective elements. However, there is still a general consensus among fans that bending as a whole grew more efficient over time. That much is evident from the movements themselves, reducing the amount of exaggerated motions to create more compact and faster strikes.
While the real world influences may be boxing and mixed martial arts, in-universe much of the reason for that paring down is due to the popularity and proliferation of pro-bending. In Legend of Korra, she her her friends, the brothers Mako and Bolin, formed a bending team, the "Fire Ferrets," and treated the audience to a masterclass on the sport's influence on their fighting styles. Offense and defense intermingle fluidly, evasive mobility is consistently maintained and the trio's styles of bending water, earth and fire are homogenized, meaning they aren't as vulnerable to mismatches against advantaged elements. But there are drawbacks as well.
In keeping with the franchise's love of subtle details, keen-eyed viewers may notice that the limitations of pro-bending affect the Fire Ferrets' bending styles just as much as its strengths do. Namely, the sport bending specializes in bursts of offense and prohibits continuously maintained attacks, defenses, grappling or building constructs. Korra learns as much the hard way when she's penalized for maintaining a stream of water for too long and freezes an elemental blast (which created a construct). The end result is that pro-benders are exceptional in the areas they're trained for, but often limit themselves from using the full range of their abilities.
Some of the moves that made Katara and Toph so superlative, for instance, were their abilities to attack from underneath their target and instantly bind them in solid ice or stone. Likewise, Aang and Azula benefitted from creating elemental constructs, like a high-speed fire wheel that could travel down a track after a target or the Avatar's famed air scooter. Such range in application not only made for some of Avatar's best moments, but allowed the fighters to effectively adapt to a variety of situations. Whereas Korra and company seldom use much more than blasts and quick shields, even outside of pro-bending, the Gaang could overcome even mismatched opponents with their creative adaptability. Creativity matters — just look at Katara versus the Sozin's Comet-empowered Azula.
Even with Azula smashing through her every defense with overwhelming power, Katara was able to think her way out of the situation by creatively using her environment and abilities. Most every final showdown with the big bads of Legend of Korra came down to displays of raw strength from the heroes winning out, and even instances like Mako electrocuting Ming-Hua were few and far between.
There are definitely advantages to what pro-bending brings to the table, but there is still plenty to learn from the more unique styles of the original cast that future Benders could learn from. A look further into Avatar's future could show an evolution that combines all such strengths, meaning fans have only seen the tip of the iceberg to what the elemental martial arts can do.