A handicap scooter is used by people who can still use their hands and aren't that paralyzed. Wheelchairs are used for people who are more weak and with possible paralysis.
For people with limited arm mobility, the wheelchair will likely have joystick control and be easier to navigate than the handlebar-style controls of a scooter. If she likes to shop, choose the wheelchair; all wheelchairs are ADA compliant. All scooters are NOT. This is a limitation in some stores.
A handicap scooter is a modern chair that one can ride on comfortably, while a wheel chair is outdated in most cases, and, is not a convenient ride for the handicapped, especially, in public places.
A motorized scooter allows ambulatory patients to ride for distances longer than they can walk. A wheelchair can also be motorized, but is generally intended for non-ambulatory patients who need the side panels and back support so they don't fall out.