Whether it is a permit or a variance will depend on the specific rules and procedures in your jurisdiction. Since DragonEggs is also in s. Florida, she may well be correct for there, but not definitively; you do need to check your own city/county procedures. I know that in MY city (Tempe, AZ), previous to the current code that allows hens at all single family residences, a variance was required (and every one submitted was approved!). If someone currently wants more hens than are allowed, or wants a rooster in a non-ag district, a variance, not a permit would be needed.
A variance is an approved deviation from the zoning code or city ordinances. How long it lasts depends on exactly how that variance is worded. For a building or structure, it lasts until such time as a certain percentage of that building or structure is remodeled or rebuilt, or until its use has been changed or abandones for one year or more. For animals it could vary depending on what the hearing officer or board of adjustments or city council agreed to (it would start at the HO, and could be appealed to the following entities in the order listed).
Permits are usually for specific activities that are already recognised or defined by specific function. Operating certain specific businesses requires a permit; adding a 2nd story to an existing house requires a use permit that would not be required if I were building a new two story house (don't ask me--seems odd to me, but that is the way it is here); this is in addition to the necessary building permit. Additional recognised permits are for apiaries, antennas more than 35' tall, etc.