If the man pays the woman for sex with dinners, a movie, flowers, etc. again there is no crime involved, at least not in civilized countries.
However, if he compensates her for her services in the form of a monetary payment, this is called prostitution and is illegal in most jurisdictions.
This is more than passing curious. Why should the form of payment play such an important, nay overwhelming, role? Money is purchasing power, more efficient than bartering goods and services, unless the recipient is going to purchase that exact combination of items in any case. For instance, for the price of a movie, dinner and flowers, the woman might prefer a pair of shoes. She could obtain the footwear, but only if she were paid in the form of money.
But the weirdness does not end there. If money changes hands, it converts an act that would be licit into a crime. The amazing thing is that the act in the two cases is identical. The only difference is the transfer of money.