Doom Safe is keeping all your skulls inside him safe from shady people. Warning: it can bite.
The difference between Piggy Bank of Doom and Doom Safe is that Doom Safe gives 1,000 which are later multiplied by all skulls' multipliers that a player has, while Piggy Bank of Doom always gives 50,000 each reset.
To clarify, let's take into consideration the following example. A player with nothing but 5 Greed Shards (x32 multiplier) and the Doom Safe, will get after reset 32,000 skulls (the x32 is applied to 1,000 skulls provided by Doom Safe).