They are military in use, and each army may have different conventions. In general, they are similar in meaning. Hope this helps :)

They mean pretty much the same thing but if you want to design the precise location where weapons and ammunitions are stored you will use the term "armory". For example: "You need to resupply your stock of ammunition, go get some at the armory". Arsenal is more general and can refer to a stock or a collection of weapons (nuclear arsenal for example) without referring to a specific location or building.

In everyday language they mean the same thing and are interchangeable. There may be differences in their use in a military context but I am not sure.