Since neutral is connected to ground how is current kept off of ground?
Current doesn't want to return to ground, but rather it wants to return to the source.  The ground isn't a great conductor and while it is in parallel with the service, the amount of current returning through the ground is so small it is effectively zero.  However, if you lose your neutral then the ground becomes the only viable return path for the current to get to the source.

I replaced a residential service and the neutral on the service drop broke on the service drop-side of the point of attachment.  The house still had power because the system was grounded and current was taking "plan B" of returning to source through the ground.  Luckily the home owners just purchased the house and hadn't moved in yet.