(1) Background: We investigated the differences in the neuropsychological profile as well as the pneumological and motor functions in two groups of patients admitted to rehabilitation who received different respiratory support during their COVID-19 infection. (2) Methods: Group-1 (n = 18; 15 male, median age 67.5) consisted of patients who received non-invasive mechanical ventilation; Group-2 (n = 19; 16 male, median age 63) consisted of patients who received invasive mechanical ventilation. All patients underwent a neuropsychological assessment including Mini-Mental State Examination (MMSE), Frontal Assessment Battery (FAB), and the Repeatable Battery for the Assessment of Neuropsychological Status (R-BANS) to evaluate the patients’ cognition. Depression and anxiety were also measured at admission and discharge to rehabilitation. (3) Results: At admission, patients impaired at MMSE were 44% in Group-1 and 5% in Group-2, while patients impaired at FAB were 88% in Group-1 and 26% in Group-2. Wilcoxon’s effect size revealed meaningful differences between groups for FAB, R-BANS global score, immediate and delayed memory, and attention-coding task, with Group-2 performing better than Group-1 across all measures. At discharge, 52% of the 25 patients re-assessed still had mild to moderate cognitive deficits, while 19% had depression and 35% had anxiety. (4) Conclusions: Patients who received oxygen therapy experienced higher levels of acute and chronic stress compared to those who benefitted from invasive mechanical ventilation. Despite patients showing a meaningful improvement at discharge, cognitive impairment persisted in a great number of patients; therefore, long-term neuropsychological follow-up and treatment for COVID-19 patients are recommended.