It is possible that a disability is the cause of a handicap. For example, if a person has a disability that prevents them from being able to move their legs, it may result in a handicap in driving.
Disabled people do not have to be handicapped, especially if they can find a way around their disability. For example, braille for the visually impaired or wheel chairs for those who cannot walk.
A disability is an inability to execute some class of movements, or pick up sensory information of some sort, or perform some cognitive function, that typical unimpaired humans are able to execute or pick up or perform. A disability may be physical, cognitive, mental, sensory, emotional, developmental or some combination of these.
A handicap is an inability to accomplish something one might want to do, that most others around one are able to accomplish. For example, reading, walking, catching a ball, or communicating.
edit The Relationship Between Disability and Handicap
The view of disability as a social construct holds that society assumes that everyone is a fully functioning, able-bodied person, which prevents the disabled from fully functioning in society, thereby creating disability.
When systems are designed thoughtfully to accommodate the needs, challenges and varying degrees of ability of different people in society, people with disabilities can fully participate in (or use) these systems. One of the major goals of the disability rights movement is to raise awareness of how systems can (and should) be designed to serve all people, not just the majority of people who happen to have no significant impairments.
For example, buildings and sidewalks that are designed to be wheelchair-accessible eliminate any handicap for people with physical disabilities (whether permanent or temporary). Closed captioning on TV lets people with hearing impairments to enjoy video programming.
edit Sensory, Intellectual or other Neurological Differences
While physical disabilities are easy to identify and appreciate, mental disabilities require the same level of thought when designing systems. Examples include sensory processing challenges that make it hard for some people to stay in very noisy environments or areas with flashing or fluorescent lights. Some kids may have attention, communication or cognitive challenges that can be mitigated by providing extra time for taking their tests. These are all examples of ways in which systems can be designed to let people overcome their disability so it does not become a handicap.
edit What is politically correct to say?
It is politically correct to say that a person has a disability and it's impolite to call someone handicapped.