It will still detect the revealed ones
A sprung trap does not stop being a trap, just because it was sprung. It still is a trap, and in some cases can be rearmed. You can even buy a hunting trap in the equipment list of the PHB (p. 190). It is clearly a trap even when not set. It's in the name. A secret door does not stop being a secret door, just because it has been discovered, it still is a secret door that when closed cannot be easily discovered by casual inspection.
If the wand was to detect only undiscovered secret doors, or armed traps, it would need to say so, and it does not.
The case Eddymage makes that a known secret door is no longer secret is problematic: unless the creators are long dead and gone, a secret door or is always known by the party that set it up or uses it. It can be known by some, and be secret to others. If you follow that logic, the wand would behave differently, depending on the knowledge of its user, because being undiscovered is not an inherent property of the trap or door.
In my experience, playing it like that — that traps spring and secret doors discovered will register for the wand — does little to lessen the wand's usefulness. We have such a wand, and nearly all of the time use it to search for secret doors in rooms or corridors that apparently have none. In the rare case where we already found one and are looking for more simple positioning yourself away from it to scan neighboring areas often does the trick.
Still, this makes the wand less useful, because you can set up a situation where one secret door or trap blocks it from discvering another. The DM may conclude that this is no fun, and decide to interpret it as only discovering secret doors and traps that are unknown to the user.