How did Buddhism and Hinduism differ in their approaches to sex work

Neither such as Buddhism nor such as Hinduism do approach anything or have views, people have, and they have as much different views as there are people, of corse.

Maybe this short quote helps in general to do not lose what is called proper attention:

"When we see Hindus worshiping Siva lingas it looks strange to us, but actually everyone in the world worships the Siva linga — i.e., they worship sex, simply that the Hindus are the only ones who are open about it. Sex is the creator of the world. The reason we're all born is because we worship the Siva linga in our hearts."

Ajaan Fuang Jotiko, in "Awareness Itself"

What's the different between different kind of selling, trading with entertainment, making trade, with ceratin objects he/she might be even up to gain.

Dhamma, of course, is directed at least to abound trading, starting with means of right lifelihood wheras those directed into the world even take Dhamma as a means of their trade. It's not wrong in it's essence, to call them participator of red light districts.

Genuine spiritual teachings cannot be separated from the manner in which they are given.

True Dhamma is like friendship: if you are being charged for it, you already know you are not getting the real thing.

Buying Dhamma, taking ungiven, stealing, and so on is like seeking for a heart of gold and then try impatiently not willing of real sacrifies satisfy one self just in redlight districts, prostitution, or abuse. Even it might satisfy shortly, it will hardly ever become ever a real, intimate and fruitful relation.

Is that what you seek for, all you can, could effort?

There have been ... prostitutes who realised the Dhamma quick, as they possible do today, if coming proper in touch.

Of the two trader, one of his/her body (kaya) and one of the Dhammakaya (body of the teachings), the second is seen as to be real nurishing much suffering and demerits for many.

One might know the joy and beneft if getting certain things free and without being not genuine and strings binding on the world.

The business of prostitution is how ever, in most Buddhist countries and people there by large not so negatively discriminated like in monotheistic countries. There is still a more "liberal" view, just another kind of trade.

This liberality in that regard is displayed by the Buddha and followed correct in the quoted sutta.

Famouse prostitutes:

• Vimala, the former courtesan: Thig 5.2

• Ambapali, Ven. Sister (formerly the courtesan): DN 16, Thig 13.1 (linked above)

There are generally five kinds of business given by the Buddha, not good for practicing lay people to get involved into:

"A lay follower should not engage in five types of business. Which five? Business in weapons, business in (human) living beings, business in meat, business in intoxicants, and business in poison."

Right Livelihood

To deal, sell or operate with sex-worker, making use of them, would be in the sphere of "business with living being, humans, salve work, so to speak, which is sure not easy to be done without getting involved in primare unskillful deeds.

Not that primaly unskillful can be traced if doing such voluntary, yet of course can involve one in conflicts brought by the buyer and of course sensual pleasure is an obstacle for the path, yet normal way of business in all sphere under lay people.

As for the seller and also buyer, he/she should be aware, this message needs to be reflected clear:

Four things befall the heedless man who lies down with the wife of another: a wealth of demerit; a lack of good sleep; third, censure; fourth, hell.

A wealth of demerit, an evil destination, & the brief delight of a fearful man with a fearful woman, & the king inflicts a harsh punishment. So no man should lie down with the wife of another.

Nirayavagga: Hell

But there is no real evidence (althought really not praiseworthy) if relation-free (or given free), "voluntary", and correct in regard of the deal, that selling ones own body is primarily to be regarded to be unskillful, not to speak of general worthy to be as "bad" blamed.

Cheating others harms, of course.

[Note: This is a gift of Dhamma, not meant for commercial purpose or other low wordily gain by means of trade and exchange]

In Buddhism, any activity based in only "lust" is unwholesome, as follows:

And what is the root of the unwholesome? Greed (lust) is a root of the unwholesome; hate is a root of the unwholesome; delusion is a root of the unwholesome. This is called the root of the unwholesome. MN 9

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Dhammapada 202. There is no fire like lust

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In Buddhism, no prostitute ever gained enlightenment. Instead, only "ex-prostitutes" (e.g. Vimala) gained enlightenment. Buddhism states a fully enlightened being is incapable of performing the sexual act (AN 9.7).

Any action that harms oneself or another is unwholesome in Buddhism (MN 61). References to prostitution in Buddhism generally include harming, for example:

Unsatisfied with his own wife, with others’ wives he’s seen in tow, corrupted too with prostitutes— that’s the way to disaster’s woe

Snp 1.6

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As they grow, the defilements possess many people; They play with fools, I believe, like demons with the mad.

In medicine they are like doctors, in business like householders, in decoration like prostitutes, in sovereignty like lords.

Cheats, frauds, false witnesses, sly: using multiple plans, they enjoy material things.

Thag 16.10

(I may add more to this answer at a later time).