A punch cutter removes a small circle from the cap. The rest of the cap stays in place around the hole. Some smokers find this neater; others find the draw too tight, especially on a cigar with a small ring gauge. A punch does not work well on a pointed figurado unless the punch is very small and the head is blunt enough for it.
Keep the punch clean, because a clogged punch leaves a plug in the cigar. A punch will not fix a bunch that was rolled too tightly; that is a draw problem caused at the rolling table.
The punch is one type of cutter, along with the guillotine and the v cut.