What is the difference between the oxidation number method and the half-reaction method?

There's no real difference between the oxidation number method and the half-reaction method.

They are just different ways of keeping track of the electrons transferred during the reaction.

The only sure-fire way to balance a redox equation is to recognize the oxidation part and the reduction part. Then you balance by making the electron loss equal the electron gain.

In the oxidation number method, you determine the oxidation numbers of all atoms. Then you multiply the atoms that have changed by small whole numbers.

You are making the total loss of electrons equal to the total gain of electrons. Then you balance the rest of the atoms.

In the half-reaction method, you determine the oxidation numbers and write two half-reactions. Then you multiply them by small whole numbers to make the loss and gain of electrons equal. Then you add the two half reactions together and balance the rest of the atoms.

In both methods, you must know what the oxidation numbers are and what they become during the reaction.

Sometimes one method is more convenient than the other method.

It's just one process and one method with variations. None of it will happen if you don't get the oxidation number of every player in the reaction.