Living in the shadow of a legend.In every person's life, there will always be enormous obstacles that must be overcome in order to achieve one's goals. Fictional characters like Falkner are not necessarily exceptions to this rule of life. He is a talented, passionate, and determined young man with lofty goals of his own and the power to reach them. Yet despite all of that, he is forced to live in the shadow of his father. He is often compared to his father (a man named Walker in the manga) in normal conversation between other trainers. There isn't much mention of how he feels, but any human being would feel the pressure of a living legend like that. I would imagine that no matter how proud he is to have such an amazing and well-liked man for a father (and he is definitely proud), he still feels the need to step out of his father's shadow and be respected for his own achievements and strength. This is probably one of the motivating factors in his drive to become not only a Gym Leader, but the greatest master of Flying-type Pokémon.
Living in his father's shadow while simultaneously swelling with pride for the man is a very human emotional contradiction. He received many things from his father - his original Pokémon and even his gym - and he is not ashamed to admit it. (aka brag about it rather adorably.) But he doesn't stop there, dedicating himself to becoming stronger and building upon what his father has given him. In the HGSS games, the first time the player battles Falkner, he only has two Pokémon. However, much later in the games, he can be challenged again and this time with a full party - one that he acquired on his own. It is realistic to assume that while the player is off getting stronger, so too are the people he's faced on his journey. The game doesn't explicitly state that, but much of the Pokémon fandom comes from things that are left unspoken anyway.
Ultimately, living in his father's shadow is something that is beneficial to Falkner - it provides a strong sense of motivation to achieve his own goals. It forced him to become the confident and strong young man he is, and will drive him to improve himself even further in the future. It is what helped to make his develop his own strong morals and values, and part of what makes him so passionate and determined. I wish I could see him accomplish his goals, but as he is not the main character of the series, and there are just so many other characters, I don't know if I will get to. I'll just have to leave it up to my imagination and hope for more.