Medicare Supplement Insurance pays whenever Medicare A and/or B pays its portion of the bill first. Often this is the hospital deductible for each hospital stay, the skilled nursing care benefit co pay, or the 20% that Medicare doesn’t cover for Part B expenses. Medicare Supplements work in tandem with Medicare itself. With a Medicare Supplement plan you can go to any provider as long as they accept Medicare and no referrals are required.  Medicare Supplements aka Medigap never cover prescriptions. Part D drug coverage would be purchased separately. This allows the consumer to customize their drug plan coverage every year. With all Medicare Supplement plans,  coverage is guaranteed for life as long as you pay your premiums.
Medicare Advantage Plans are completely private plans which actually replace Medicare itself  (it doesn’t supplement Medicare). Your Medicare card is “put to sleep” while you are on a Medicare Advantage Plan. Medicare Advantage Plans usually cover prescription drug cost(known as Part D) but not always if you choose a plan without Part D. Medicare Advantage plans also offer additional benefits that aren’t covered by Medicare itself, such as Dental, Vision, hearing, gym membership etc. Typically you have a lower monthly premium with Medicare Advantage but more co-pays or co insurance until an out of pocket limit is satisfied.  Also, many Medicare Advantage plans are HMO plans which require you to stay in network except for emergency care. The coverage is not guaranteed for life as plans contract with Medicare on a year to year basis. Benefits can also change from year to year where Medigap plan benefits never change unless the policy holder changes their plan on their own.
Someone on Medicare Part A and B can always switch from Original Medicare to a Medicare Advantage between Oct 15th and Dec 7th of each year. However, they are limits to going back to a Medicare Supplement plan later on due to pre existing health issues.