I have looked up if it is legal to buy a tank, and I found that you need a FDDL (Federal Destructive Device License). Is that all one really needs, or would one need to get a gun permit as well?
As far as I can tell, there is no state or federal law which prohibits a private individual from owning a decommissioned military tank.
Most public highways and roads have weight limits, and many tanks would exceed them. Most tanks, or at least most older tanks, are not in any case "street legal" not having required headlights, brake lights, air bags, and other safety devices. Treads must be modified to avoid road damage. None of this would be relevant if the tank was kept on private land and not used on public roads or streets.
If somehow the main gun or a mounted machine gun were still in place, and not disabled, permits would be required that are almost impossible to obtain. Specifically:
Code section 4-401 which provides that:
(c) Machine gun. "Machine gun" means a loaded or unloaded weapon that is capable of automatically discharging more than one shot or bullet from a magazine by a single function of the firing device.
(a) Evidence of possession. The presence of a machine gun in a room, boat, or vehicle is evidence of the possession or use of the machine gun by each person occupying the room, boat, or vehicle.
(c) Registration of possession.
(1) A person who acquires a machine gun shall register the machine gun with the Secretary of State Police:
(i) within 24 hours after acquiring the machine gun; and
(ii) in each succeeding year during the month of May.
(b) Destructive device.
(1) "Destructive device" means explosive material, incendiary material, or toxic material that is:
(i) combined with a delivery or detonating apparatus so as to be capable of inflicting injury to persons or damage to property; or
(ii) deliberately modified, containerized, or otherwise equipped with a special delivery, activation, or detonation component that gives the material destructive characteristics of a military ordnance.
(2) "Destructive device" includes a bomb, grenade, mine, shell, missile, flamethrower, poison gas, Molotov cocktail, pipe bomb, and petroleum-soaked ammonium nitrate
(a) Prohibited. A person may not knowingly:
(1) manufacture, transport, possess, control, store, sell, distribute, or use a destructive device; or
(2) possess explosive material, incendiary material, or toxic material with intent to create a destructive device.
You can own a properly de-militarized tank1. You can't drive it on public roads2. You can't have any of the tank cannons or machine guns3&4. Properly demilitarized means it's a metal shell without an engine.4
1: §13a KrWaffKontrG (Kriegswaffenkontrollgesetz - Law on the control of weapons for war) defines properly De-militarized as permanently disabled and unable to be used in military ways. Any gun-looking things that remain have to be disabled so thoroughly that they can never be reactivated in any way again - this means gun barrels generally are to be removed or utterly destroyed. But the armor is to be removed in parts too, to no longer qualify as a weapon of war. So far so peachy, yes? well, laws see to that...
2: § 19 Absatz 2a StVZO (Rules to obtaining a vehicle's operation allowance) says, that vehicles that are in their design military vehicles lose their operation allowance once they leave military service. As a result, you can't register the tank for road traffic, and you can't get a license plate unless you manage to get the tank through the whole process of applying for a new road vehicle. Without a license plate, you can't drive a tank on any public road. So for now, you can only drive your tank on private property with the allowance of the owner only. In general, the only tracked vehicles that even can get a road license are construction vehicles. But... there is more!
3: The KrWaffKontrG appendix lists items that are controlled. Such as part B category V, including any and all machine guns (point 29), tank cannons (point 31), and part V category IV, especially point 24 and 25 for battle tanks and armored vehicles. These all are weapons of war that require special licensing to make and import. So far no trouble there? Well... no, those categories are your problem:
4: There are rules upon handling these items in the shape of the KrWaffUnbrUmgV (rules how weapons of war are to be demilitarized and handled afterwards), especially §4: this bans the ownership of armored vehicles and tanks as soon as they have a working engine. As a result, you need a special license to obtain, own and operate an armored vehicle with an engine (appendix point 25)3, and you can't get that license for battle tanks (appendix point 24)3. And category V3 are banned too, though here some exceptions apply.
If you want to have a dozen-ton paperweight... you can buy a hunk of scrap that doesn't have an engine in Germany. But you need a special license to even own an armored car.
When they changed the laws the last time, the lawmakers did mention in their debates Michael Fröhlich, a car collector who drove his tank licensed in Cologne through Düsseldorf between 1997 and 2000. While he arguably was the last private citizen to get a license for his tank, he wasn't the first to drive a tank through German towns: Herbert Mittländer had driven a tank through Frankfurt in 1976, after he had replaced a large chunk of the front with aluminium and added the needed lights... only he lost his operation license rather quick again - for lack of a crumbling zone.
You can buy a De-Militarized (Decommissioned) Tank but there would be certain rules you'd have to follow. You can't drive it like an everyday vehicle. You might be allowed to drive it in public parades. You can't drive it on public roads probably because of the weight of the tank.
I too have wondered this question and actually talked to by district representative about this and this is what I was told.