My son got his junior license in January of this year when he was 11. He has since turned 12. Does he need to get a different license in order to purchase a deer tag and hunt big game?

It depends on which license your son has.  If he bought a Jr. Small Game license in January when he was 11, but is now 12 and wants to hunt deer, he will need to purchase a Jr. Hunting License in order to purchase a deer tag.