The only example I know is Charlie Fefferman and his first student.

Advisor: Fefferman, born April 18, 1949. Year of PhD 1969.
Student: Antonio Córdoba Barba, born January 12, 1949. Year of PhD 1974.

Difference in age: 3 months and change.

Trivia: that is the only example I know also where both father and son studied under the same PhD advisor.