Post-COVID Syndrome has emerged as a significant public health concern worldwide with increasing evidence to suggest that individuals who have had an acute COVID-19 infection report lingering memory and attention difficulties, even in individuals who have fully recovered and no longer experiencing symptoms of COVID-19. The present study sought to investigate the profile of objective and subjective cognitive difficulties in people who have Post-COVID Syndrome, people who have fully recovered from an acute COVID infection and people who have never had COVID-19. We further sought to explore the extent to which self-reported fatigue and stress are related to subjective and objective cognitive difficulties. 162 participants including 50 people living with Post-COVID Syndrome, 59 people who have had COVID-19 but have fully recovered and 53 people who have never experienced symptoms of COVID-19 and had never tested positive for COVID-19 were recruited from Academic Prolific to complete a series of online questionnaires and neurocognitive tasks. Subjective cognitive function was measured using the Cognitive Failures Questionnaire and objective cognitive function was measured using the Cognitron cognitive test battery. We found that objective and subjective measures of cognitive function were not significantly related, suggesting that self-reports of "brain fog" are not reflecting objectively measured cognitive dysfunction. A MANOVA revealed that subjective cognitive deficits were driven by heightened perceived stress and fatigue and not significantly related to COVID-19 status. Objective cognitive function, however, was significantly related to perceived stress and COVID status whereby we observed significant objective cognitive deficits in people who have been exposed to an acute COVID-19 infection regardless of whether they had Post-COVID Syndrome or had fully recovered, as compared to people who had never had COVID-19. This suggests that an acute infection can have long term effects on cognitive function, even without persistent COVID-19 symptoms. Encouragingly, objective cognitive function was significantly associated with time since initial infection showing that cognitive deficits improved over time for people who had recovered from COVID-19. However, we did not observe the same improvement in individuals with Post-COVID Syndrome and observed that cognitive dysfunction was significantly related to the number of neurological symptoms presently experienced. These results add to the accumulating literature that COVID-19 is associated with significant cognitive difficulties following a COVID-19 infection, which appear to improve over time for those who have recovered from COVID-19 yet persist in people living with Post-COVID Syndrome.