The music and the way it is said.
The Gradual is two psalm verses, set to a very complex and melismatic melody, often (especially those in mode V) with a change of key part way through. The schola intones the first word or two, and the choir sing the first verse, then schola only usually chants the second verse which is often even more complex. This is not to be confused with the “Gradual Psalms” which are psalms 119-133 and were the hymns of the pilgrims to Jerusalem. The Gradual, in the Mass, can be from any psalm, not just the Gradual psalms.
The responsorial psalm is just a psalm or part of a psalm with an antiphon between strophes. The cantor or lector usually recites/chants the antiphon, the assembly repeats the antiphon, then the cantor or lector recites/chants the psalm with the assembly responding between strophes.
Both are licit options for the Ordinary Form Mass. Where I attend Mass (a Benedictine monastery) in the Ordinary Form, the Gradual is used, and is chanted in Latin to the traditional melodies in the Graduale Romanum.
In both cases, the psalm fills the interval between the first and second readings.
