How is an "assault" different from "battery" in everyday English?

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.