Does a fully charged battery weigh more or less than a battery that has been depleted?

No change. It’s a chemical reaction, and electrons are transferred from one chemical to another, but it’s a closed system. I have seen arguments that mass will decrease because of the special theory of relativity, but it’s less than trivial.

Another way to think of it is as a water turbine in a dam. The water weighs the same above the turbine as below, but work is done and power is produced.

I like @Caravanfan‘s answers. A sealed battery will remain the same. By charging it you just moved some electrons or ions from one side of the battery to the other.
You can also think of it like compressing a spring. The compressed spring has stored potential energy yet it weighs the same as a released spring.

For an unsealed lead acid battery, the answer is slightly different because of a “technicality”. A small amount of Hydrogen and oxygen gas is released when a lead acid battery is charging. You can see the bubbles on the plates if you take the caps off and look inside. Occasionally you hear about a battery exploding because someone had an open flame nearby and the Hydrogen ignited.
The acid water mix in the battery that has been broken down needs to be replaced occasionally to keep it at the proper level. That is why you need to check your car battery and fill it with distilled water.
How much hydrogen is released while charging and how much does it weigh? Think about how often you filled your car/motorcycle/tractor batteries – not very often but you do need to do it occasionally The typical lead acid battery will discharge 0.016 cubic feet per amp hour per cell. You can do the math if you want to take it further,
For factories that use battery operated forklifts the hydrogen released when charging can be a real safety problem .
Here is a Battery Hydrogen Concentration calculator supplied by Cisco, Inc that determines the hydrogen loss during charging and the concentration in the room.

I have very accurate scales at my disposal and use them for all sorts of odd applications.
I have a spring piston operated pellet gun Beeman P17 that is used for various purposes around the house.
It is not CO2 powered. It is operated by cocking a lever that compresses the air in the main power piston. The gun weighs about 795 grams but when it is cocked it weighs 0.07 grams more than uncocked! Why? Because the compressed air is more dense than ambient air and since the gun is not a sealed system, ambient air has has filled in the void.
So I can tell if the gun is cocked just by weighing it.
Physics is fun!

A fully charged car battery is full of sulfuric acid. A fully discharged battery is full of water. I can’t say for sure but it seems the specific gravity of the acid would be less than that of water.