The advantage of June 1 cuts is you spread the cap hit out over 2 years. And you can cut as many players as you like after June 1.
However, you can also cut a maximum of two players as early as today and designate them as June 1st cuts. There is no cap relief for the team until June 1; all it does is allow the released player an early start on free agency. It is a benefit for the player only. There is no benefit for the team.