Background: The SARS-CoV-2 pandemic has significantly affected the pediatric population. Long-term sequelae (Long COVID-19) may particularly involve the central nervous system, with possible effects on psychological well-being and quality of life (QoL), aspects that were already influenced by the restrictive measures and general social impact of the pandemic. Methods: We conducted a cross-sectional survey that aims at investigating the neuropsychological effects and the QoL impairment of SARS-CoV-2 on a cohort of children and adolescents in the Abruzzo region (Italy). A questionnaire was submitted to caregivers with the help of the PEDIATOTEM platform. A control group of healthy subjects was also included to distinguish between the effects of infection from the general influence of the pandemic. Results: A total of 569 subjects responded: 396 COVID-19 patients (99 of whom had Long COVID-19) and 111 controls. After the pandemic, when compared with the COVID-19 group, the controls reported significantly increased appetite, sleeping habits, and time spent remotely with friends and a reduction in physical activity and time spent in person with friends. A significant higher rate of controls asked for psychological/medical support for emotional problems. On the other hand, the Long COVID-19 group showed more fatigue and emotional instability with respect to non-Long-COVID-19 subjects. No differences in QoL results (EuroQOL) were found between the COVID-19 patients and controls, while the Long-COVID-19 subgroup showed significantly higher rates of pain/discomfort and mood instability, as confirmed by the analysis of variation of responses from the pre-COVID-19 to the post-COVID-19 period. Conclusions: Among COVID-19 patients, neuropsychological and QoL impairment was more evident in the Long COVID-19 subgroup, although emotional and relational issues were also reported by uninfected patients, with a growing request for specialist support as a possible consequence of social restriction.