Medicare was created to address the fact that older citizens have medical bills significantly higher than the rest of the population while they have less money to pay those bills.  Medicare is an entitlement program.  You are entitled to Medicare because you or your spouse paid for it through social security taxes.

Medicaid is a Federal program for low income people.  There are separate eligibility requirements for each program.  Being eligible for one does not necessarily mean you are eligible for the other. If you qualify for both, Medicaid will pay for most Medicare Part A and B premiums, deductibles and co-payments.