The lifespan of people with cognitive disabilities (ID) has increased significantly, but the cognitive aspects together with the functional ones comparing normal aging and those with intellectual disabilities had not been previously studied. Objective: This study analyzed the cognitive and functional differences in older adults aging with ID (and with DS), compared with their peers without disabilities, in order to identify the most adapted interventions. Methodology: This study evaluated the outcome variables of MEC, Set-Test, Barthel, Lawton–Brody, and Tinetti with 247 participants: 146 without ID and 101 ID (29 with DS and 72 without DS). Results: At the cognitive level, older people with ID presented lower scores both in MEC (p < 0.01), globally and in each cognitive domain (except in short-term memory), and in verbal fluency (Set-Test) than older people without ID; however, the diagnosis of cognitive impairment and dementia is higher in people without ID. At the functional level, there are no differences in ABDL, but there are in AIDL and Tinetti (p < 0.01), where participants without ID obtain higher scores. The most frequent pathologies in people with ID were obesity and epilepsy. Conclusions: The lower cognitive and functional performance in ID is associated with the disability itself, the low educational level, the neurocognitive underdiagnosis, and the use of poorly adapted assessment tools. The cognitive and functional results indicated the importance of interventions adapted to the characteristics of this population, in their aging process.