The pressure is the same inside and outside, but the helium is lighter so the balloon floats. From the ideal gas law PV=NRT and plugging in the constants you get the well known rule that at standard temperature and pressure, the volume of 1 mole of gas is about 22.7 liters. It's weight depends solely on the gas's molecular weight.

The molecular weight of helium is 4 while for air (a mix of N2 and O2) it's a little over 28. That should let you calculate the lifting force for a given size of balloon.

The very short answer is that gas pressure is mostly proportional to the amount of particles per volume.

So a balloon filled with helium has X particles per cubic cm, while the air around it has the same amount (instead of getting crushed). But because helium is a lot lighter per particle than standard air, this makes the balloon lighter than air, and like trying to push an air-filled balloon underwater, this helium-filled balloon floats to the higher layers of air, until other smaller forces also start to matter and the balance is restored.

So a "vacuum-filled" balloon has nothing to give counter-pressure, but a balloon filled with helium definitely does.

Not my area of expertise but density is the wrong concept to be looking into remaining equal, its pressure. The amount of helium gas you pump in to the balloon creates a certain outward pressure that’s at equilibrium with outside air pressure, that’s why it doesn’t collapse (if you were to place the balloon in a vacuum then it would expand and probably burst, that’s usually what happens when a child loses one and it escapes higher up in the atmosphere to lower pressures)

The atmospheric pressure simply isn’t strong enough to overcome the outward pressure.

Bonus point: density is actually the reason why a helium balloon floats, its less dense than air so it wants to float up. Much like a rubber duck or ice cube in water.