Which is greater: atmospheric pressure or gas pressure inside the balloon? Why?

The air pressure inside the balloon is higher, because it is inflated from the inside. The air exerts more pressure on the wall from the inside than from the outside (atmospheric), so the ball swells until the elastic forces of its rubber wall compensate for the difference between the pressure inside and outside and equilibrium occurs. If you inflate the balloon harder, it will burst, because the rubber wall will not withstand more pressure. The air pressure inside and outside the ball will be the same if the air is deflated completely from the ball.