Among the many differences between the two Avatars, Korra was able to bend in the spirit world in The Legend of Korra while Aang couldn't in Avatar: The Last Airbender. Thousands of years before the events of the original series, humans and spirits lived in harmony - that is, until a man named Wan severed the connection between the spirits of light and darkness, Raava and Vaatu, respectively. In doing so, he inadvertently caused chaos in the world, and the only solution was to separate the realms.
And so by binding with the spirit Raava, Wan became the first Avatar. Although humans could bend the elements, albeit only one at a time, Wan was able to harness Raava's power to bend all four elements. And using his newfound powers, Wan defeated Vaatu and closed the portals to the spirit world, thereby disconnecting the two realms for almost 10,000 years - until Korra reopened the portals. Wan's actions in turn created the era of the Avatar, in which the Avatar was the bridge between the two worlds. For a time, the Avatar was really the only person who would cross over - but when he or she did so, they didn't have their powers to rely on.
One of the rules of Avatar's spirit world is that anyone who enters can't use their bending powers... unless they enter through one of the spirit portals. By meditating their way into the spirit world, people's spirits travel into that realm but their physical bodies remain behind in the real world. However, physically entering the spirit world through one of the portals means they can still bend there. Aang never physically entered the spirit world, so he could never bend in there, but Korra could since she - and her friends - did physically enter that area.
Interestingly, there's more to Avatar's rules about bending in the spirit world beyond physically being there. While benders can't use their powers by meditating into the spirit world, Avatars actually are able to use one power - energybending - without being there in their material forms. Energybending is the original bending form; before people - and the animals before them - could bend the four elements, the lion turtles would first bend the energy inside themselves. It's so powerful that a person using it could grant and remove one's bending.
One lion turtle gifted Aang with energybending, which he then used to take away Fire Lord Ozai's bending in the series finale rather than kill him. Korra later used energybending for the opposite reason, to give back people's bending powers after the villain Amon took them away with bloodbending. Although it's never been explained explicitly, it's plausible that the reason energybending is possible in the spirit world without the bender's body is because the power relies entirely on the spirit itself. All one needs is their spirit - and the Avatar's spirit is most powerful in the spirit world. Korra proved this in Legend of Korra season 4, when she rescued people trapped in the spirit world by energybending.