The difference is that whereas puny usually means physically weak, puisne means not so powerful.
It's used for people who have too much authority and dignity to have sand kicked over them when sunbathing. Particularly judges.
There are times when calling someone in great authority puisne can be extremely refreshing.
Best not say it out loud, though.
Thing Not To Tell Someone They Are Today: puisne. This word comes from the Old French puisné, born later, from puis, at a later date, and né, born, from the Latin nascī.