Susanne asks a valid question (both words are used in both contexts), although I’m going to imagine you’re asking about the words in a legislative context. For one thing, that’s what your Chinese terms seem to indicate, and more importantly, it gives me an excuse to mention this funny classic video from “Schoolhouse Rock” explains how a bill becomes a law.
https://www.youtube.com/watch?v=Otbml6WIQPo
As far as the difference between bill and act, the McMillan defines a bill as “a written document containing a proposal for a new law,” and an act (or Act) as “a law passed by a country’s government.” In other words, a bill is a formal, written proposal for a new law. If and when the bill is passed in accordance with the legislative procedures of the country in question, the bill becomes a law. In this context, “act” is synonymous with “law,” in the countable sense — a law passed by the legislature. I’m not a lawyer, of course, and I didn’t consult a legal dictionary, so this is just to give you an idea of how educated native speakers understand these terms.