Incompetent implies inability ("not“not up to the job"job”) whereas inept could be a judgement on something not being done well on a particular occasion. For example it would not be a contradiction to say: "No one disputes Charlie's competence in managing his employees, but his handling of the dispute between John and James was manifestly inept."
“No one disputes Charlie’s competence in managing his employees, but his handling of the dispute between John and James was manifestly inept.”
