This work introduces a new socially assistive robot termed MARIA T21 (meaning “Mobile Autonomous Robot for Interaction with Autistics”, with the addition of the acronym T21, meaning “Trisomy 21”, which is used to designate individuals with Down syndrome). This new robot is used in psychomotor therapies for children with Down syndrome (contributing to improve their proprioception, postural balance, and gait) as well as in psychosocial and cognitive therapies for children with autism spectrum disorder. The robot uses, as a novelty, an embedded mini-video projector able to project Serious Games on the floor or tables to make already-established therapies funnier to these children, thus creating a motivating and facilitating effect for both children and therapists. The Serious Games were developed in Python through the library Pygame, considering theoretical bases of behavioral psychology for these children, which are integrated into the robot through the robot operating system (ROS). Encouraging results from the child–robot interaction are shown, according to outcomes obtained from the application of the Goal Attainment Scale. Regarding the Serious Games, they were considered suitable based on both the “Guidelines for Game Design of Serious Games for Children” and the “Evaluation of the Psychological Bases” used during the games’ development. Thus, this pilot study seeks to demonstrate that the use of a robot as a therapeutic tool together with the concept of Serious Games is an innovative and promising tool to help health professionals in conducting therapies with children with autistic spectrum disorder and Down syndrome. Due to health issues imposed by the COVID-19 pandemic, the sample of children was limited to eight children (one child with typical development, one with Trisomy 21, both female, and six children with ASD, one girl and five boys), from 4 to 9 years of age. For the non-typically developing children, the inclusion criterion was the existence of a conclusive diagnosis and fulfillment of at least 1 year of therapy. The protocol was carried out in an infant psychotherapy room with three video cameras, supervised by a group of researchers and a therapist. The experiments were separated into four steps: The first stage was composed of a robot introduction followed by an approximation between robot and child to establish eye contact and assess proxemics and interaction between child/robot. In the second stage, the robot projected Serious Games on the floor, and emitted verbal commands, seeking to evaluate the child’s susceptibility to perform the proposed tasks. In the third stage, the games were performed for a certain time, with the robot sending messages of positive reinforcement to encourage the child to accomplish the game. Finally, in the fourth stage, the robot finished the games and said goodbye to the child, using messages aiming to build a closer relationship with the child.