'Distinct' and 'Different are nuances of the same notion of difference. Different simply means "which are not the same". Distinct, however, refers to a sharper difference, something that is very noticeable, which stands out. When you want to emphasize the difference between two things/persons/elements, use the word "distinct" (out of the pile of various keys, there was one distinct key that served to open the door to the room)
'Distinctive' refers to features or qualities that serve to tell something apart (that bird's distinctive traits are its red feathers)
A crowd may be made of people who are different, a distinct person, is one that stands out from a crowd.
'Distinguished' means having an air of dignity, manners, a noble air, ("a distinguished gentleman), eminence, or being famous ("a distinguished scholar").