In legal parlance, the word "assault" historically means an attempted battery (battery being defined as below) or an intentional frightening of another person. No contact is necessary. On the other hand, battery is when the unwanted contact actually takes place.
Does everyday usage of these words tacitly acknowledge this difference?
In everyday usage, I'd say that assault is commonly used, and means 'to attack', which is slightly different to the legal meaning which is the threat of an attack (battery), combined with the a demonstration of the means to attack.
Battery is less commonly used outside the law, however, in the UK at least, the verb to batter is still commonly used:
In the playground:
Cough up or I'll batter you.
= Please give me your dinner money, or I'll be inclined to commit battery
In the pub:
Q: How did the game go?
A: I absolutely battered him!
= I won by quite a margin!
In the U.S. at least, the word battery is so rarely used outside the legal phrase assault and battery that a listener would be pretty much guaranteed to assume it meant an electrical battery unless it was specifically disambiguated by context. So I think perhaps the question is misguided, as I can't say with any certainty that there is in fact an "everyday usage" of both words. Assault on its own, however, does tend to be used as shorthand for sexual assault, whereas "battery" is wholly separate from that connotation.
Maybe Genesis didn't make it to the US? Being in the lyrics (and title) of "Robbery, Assault and Battery" makes battery a pretty common word. I agree that context is important. I've always thought of assault as being a single instance with battery being much more serious, i.e. a punch thrown after an argument would be an example of assault, but someone black and blue after repeated assaults would have been subject to battery.
In legal terms, you can have a battery without assault, and an assault without battery. I'm not sure I would characterize "assault" as an attempted battery; an assault can be committed without any subjective intention to harm, as in pointing a gun at another. A battery can be committed without an assault as in the case where an individual is struck from behind. The key to (legal) assault is apprehension at an imminently pending act of harm towards oneself. Take away the apprehension and the act is no longer an assault though it may ripen into a battery.
