The COVID-19 outbreak has disrupted the daily routine of the population worldwide, including autistic people and their caregivers, with severe consequences on mental health. On one hand, the reduced social contacts and the interruption of outpatient and daycare services during the lockdown have represented a real challenge for autistic people and their caregivers. On the other hand, confinement has allowed individuals to spend more time pursuing their interests and stay home with their family members without feeling the pressure of social expectations. The present study aimed to compare the levels of personal wellbeing, family distress, insomnia, and resilience between caregivers of autistic people and caregivers of people with other neurodevelopmental, psychiatric, or relational disabilities. A web survey was completed by 383 participants, of which 141 were primary caregivers of autistic people. We did not find any significant difference between caregivers of autistic and non-autistic people in any of the considered psychological variables. Lower age of the autistic family member and lower resilience levels were significantly associated with higher individual distress in the group of caregivers of autistic people. Our findings do not corroborate the hypothesis that caregivers of autistic individuals have had more severe consequences than other caregivers during the lockdown. However, they confirm the importance of promoting resilient coping strategies in autistic people and their caregivers.