One possible reason could be their different approaches to life post - war. Aang is the Avatar and has a whole world to look after, while Katara has her own tribe and personal goals. Their busy schedules might pull them apart. For example, Aang might be constantly traveling to different nations for diplomatic and Avatar - related duties, leaving Katara feeling lonely. Also, they might have different ideas on how to handle future threats. Aang may prefer a more pacifist approach which Katara might not always agree with, especially if it endangers the people she loves.
Jealousy might play a role. Aang is a well - known figure around the world as the Avatar. There could be many girls or guys (depending on how you view their relationships) who admire him. Katara might start to feel jealous of the attention he gets. Similarly, Katara is a strong and beautiful waterbender, and there could be others vying for her attention, which Aang may not be comfortable with.