What is the ruling on a woman looking at a man, with or without desire, and what is the difference between that and a man looking at a woman?

It is forbidden for a man to look at a non-mahram woman with lust or fear of temptation, by agreement [of scholars]. It is also forbidden for a man to look at the face and hands of a non-mahram woman according to the Shafi'is and Hanbalis, and this is the correct view.

As for a woman looking at a non-mahram man: The Hanafis permit her to look at anything except what is between the navel and the knee if she is safe from temptation. The Malikis permit her to look at the face and extremities if she is safe from temptation. The Shafi'is, however, deem it impermissible for her to look at the 'awrah or anything else except out of necessity, citing the generality of the verse on "lowering the gaze" and the hadith of Umm Salamah. The Hanbalis permit a woman to look at what is not 'awrah of a non-mahram man.

The Shafi'is do not differentiate between a man looking at a woman and a woman looking at a man. Thus, it is forbidden for each of them to look at the entire body of the other, and this is the more cautious approach. If the looking is with lust, then it is not permissible for either of them to look at the other.