An obvious example of this is pop music. Often following the verse chorus form and also different genres having different constraints in terms of instruments used.
"same tune, wildly different lyrics" is as old as rocks, too.
some trivial examples involve roughly every hymn, "god bless the queen"/"my country 'tis of thee", "john brown's body"/"battle hymn of the republic", "to anacreon in heav'n"/"the star spangled banner" ....
The other big case are missals: same words, vastly different settings. You know the lyrics are the Mass, but how they're sung is where you show off.
some trivial examples involve roughly every hymn, "god bless the queen"/"my country 'tis of thee", "john brown's body"/"battle hymn of the republic", "to anacreon in heav'n"/"the star spangled banner" ....
The other big case are missals: same words, vastly different settings. You know the lyrics are the Mass, but how they're sung is where you show off.