In the US, OSHA enforces safety laws...they are "the law". (maybe you already know this...)
ANSI is an independent organization that holds standards developed by those in the industries, not laws forced upon the industry by the government. ANSI covers a very wide array of industries and it is not just safety. In our industry, for example, the ANSI A300 define standard tree care practices (including pruning, cabling/bracing, lightning protection, fertilization, risk assessment, etc, etc...).
For most fields/professions/industries, OSHA has their own set of defined standards that must be followed. The do NOT have a standard for the tree care industry (an industry with one of the highest injury rates). In absence of their own, OSHA is supposed to rely on "industry standard practices". That's vague, and can probably be defined however the inspector chooses to define it (which may or may not hold up in court if it goes there). However, there is a pretty good argument for the ANSI Z133 to be that standard that they use. It is understood that if an employer were following the Z133 to a letter, they'd have a pretty good defense against claims they are non-compliant...but that isn't spelled out explicitly anywhere in law.
(***As I understand it...subject to correction by somebody who knows more than me!!!***)