Killua cries because he remembers that his own weakness of fighting with the aim of running away will cause him to leave his best friend, Gon, to die.
Killua was born into the famous Zoldyck family of assassins. His family has already decided that he would be an assassin, and put him through harsh training for that purpose, right from childhood.
After meeting Gon during the Hunter exam, Killua decided that he wanted to be Gon's friend and lead a normal life, not be an assassin. As they spent time together during the later arcs, Gon's friendship became the most important thing for Killua. It was also his source of hope for escaping from the life his family had pre-decided for him.
Shortly before this incident in the question, Biscuit points out that Killua's weakness is he fights with the aim of running away, and further notes that it would cause Killua to leave Gon to die some day. This wasn't a mean taunt by an enemy, but an assessment of his weakness by his Nen teacher, and he probably realized that it was true.
It would have been immensely painful for him to realize that his weakness would cause him to leave Gon to die, Gon being his best friend, the person who changed his life and gave him hope of leading a normal life. For that reason, he decided to protect Gon for 30 days, and then leave him forever.
When Gon says that "being weak is so painful", it resonates perfectly with Killua's own feelings, and so he cries too.