(1) Background: Persistent COVID is characterized by the presence of fatigue, mental fog, and sleep problems, among others. We aimed to study cognitive abilities (attention, executive functions, memory, language) and psychological and emotional factors in a group of participants of the population with persistent COVID-19 and asymptomatic or non-COVID-19-infected patients; (2) Methods: A total of 86 participants aged 18 to 66 years (X = 46.76) took part in the study, with 57 individuals (66.27%) in the experimental group and 29 (33.73%) in the control group. A comprehensive assessment included neuropsychological evaluations, evaluations of anxious and depressive symptomatology, assessments of the impact of fatigue, sleep quality, memory failures in daily life, and the perceived general health status of the participants; (3) Results: significant differences between groups were found in incidental learning within the Key Numbers task (U = 462.5; p = 0.001; p = 0.022) and in the Direct Digit Span (U = 562; p = 0.022), but not in the Inverse Digit Span (U = 632.5; p = 0.105). Differences were also observed in the prospective memory task of the Rivermead Prospective Memory Tasks (from the Rivermead Behavioural Memory Test) in the recall of quotations (U = 610; p = 0.020) as well as in the recall of objects (U = 681.5; p = 0.032). Concerning the task of verbal fluency, significant differences were found for both phonological cues (p- and s-) (t = -2.190; p = 0.031) and semantic cues (animals) (t = -2.277; p = 0.025). In terms of the psychological impact assessment, significant differences were found in the emotional impact across all variables studied (fatigue, quality of sleep, memory lapses, and the perceived general health status), except for quality of life; (4) Conclusions: Our results suggest that the sequelae derived from persistent COVID may have an impact on people's lives, with higher levels of anxiety and depression, worse sleep quality, a greater number of subjective memory complaints, and a greater feeling of fatigue and impact on quality of life. Furthermore, poorer performance was observed in memory and verbal fluency.