A quick summary - - The process is the same whether you are filing for SSDI or the Supplemental Security Income program.
This was an excellent explanation of the two programs. There are some other significant differences in how the programs work once someone is approved. (The numbers that follow are based on my recollection, so they may not be exact but should be close.)
SSI pays a very low amount - Many people get less than $800 a month. And the recipient cannot have any significant assets - no more than $2,000. If they have a car, it's got to be worth less than that amount. It really is designed to be a "safety net" to help those who otherwise would be unable to provide food and shelter for themselves because of disability. Of course, in many places that's not enough to pay rent and buy much food, much less anything else and the individual's financial records are monitored, I believe. The one additional benefit is that in many states, qualifying for SSI also automatically ensures that you will also qualify for your state's medical assistance program for the disabled, giving you some medical coverage.
Because SSDI is based on someone's earnings they payment is higher - like Social Security retirement payments, it's linked to how much someone paid in. But of course, it's still less than most people would make in a paying job. It also doesn't have the stringent limits on assets, so recipients have more flexibility in managing their lives.
The Wall Street Journal had an interesting article today on how rapidly disability has grown, most recently likely because the recession has pushed people who might been able to work through their disability into the SSDI system because jobs just weren't available. There's also an article about the huge backlog that's existed in the system, based on needing doctors to review all the applicant records. There's a battle between the doctors doing the review, who believe it's important to have a medical background in the disability issue being assessed, rather than just being an MD, and the SS system which disagrees and wants to push the review faster. It's also got an interesting map that shows how some parts of the US - primarily in the SE, many more applicants get approved.