In some cases they are synonyms. Distinct can mean different when something is clearly or unquestionably different. The difference is easy to see, obvious, easy to describe.
"At first the scientists thought the two birds belonged to the same family, but DNA tests showed that they belong to two distinct species." Their difference is unquestionable now.
"The prime minister and the president perform two distinct roles."
It's even possible to say "distinctly different."
Distinct also means clear or noticeable. "The was a distinct improvement in the company's sales after reorganization."