What is difference between the terms aging, old age, and elderly?

The American Psychological Association (1) states in Section 2.17 Age, (p 69):
“Elderly is not acceptable as a noun and is considered pejorative by some as an adjective. Older person is preferred. Age groups may also be described with adjectives: gerontologists may prefer to use combination terms for older age groups (young-old, old-old, very old, and oldest old), which should be used only as adjectives. Dementia is preferred to senility; senile dementia of the Alzheimer’s type is an accepted term.
The American Medical Association states in Inclusive Language Section, 9.10.3 (p 268):
Age.–Discrimination based on age is ageism, usually relevant to older persons. Avoid using age descriptors as nouns because of the tendency to stereotype a particular group as having a common set of characteristics. While in general the phrase the elderly should be avoided, use of the elderly may be appropriate (as in the impact of Medicare cuts on the elderly, for example). Otherwise terms such as older person, older people, elderly patients, geriatric patients, older patients, aging adult, or the older population are preferred.
Note: In studies that involve humans, age should always be given specifically. Researchers in geriatrics may use defined terms for older age groups, eg, young-old (usually defined as 60 or 65 to 70 or so years) and old-old (80 years and older).
In an ASA Connection newsletter (3), the following responses were reported for the prior month’s Question: What terms do you think are appropriate when referring to people ages 65-plus? Older adults, 80%; elders, 41%; seniors, 33%; senior citizens, 11%; elderly, 10%. Note: Total is >100% because respondents could select >1 answer. ASA Connection provides updates on events in aging, research and policy developments, and innovative practices. The newsletter is distributed monthly to members of the American Society on Aging and other professionals in the field of aging.
“Senior” is considered passé, especially by today’s 78 million baby boomers. Recently some YMCA organizations have begun to refer to the older adults who participate in their fitness or physical activity programs as AOAs, ie, Active Older Adults.