The non-legalize difference is as follows:
1) An assault means you were not actually hurt, but scared. If you pull a gun someone, but do not shoot it is an assault. The person is not physically harmed, but they are certainly scared of being hurt.
2) Battery is a harmful or offensive touching. If the person shoots and the bullet hits you, then it is battery among other things; but since there was contact, it is battery.
Very short explanation is assault means there is nothing touches you or hurts you, battery means something touches or hurts you.
Here is a brief article explaining the difference.
From another reader:
Assault is the willful and unlawful attempt, coupled with the present ability, to do harm upon the person of another. Battery is the willful and unlawful use of force upon the person of another.
If I threaten to punch you, I have committed an assault. If I punch you, I have committed battery.