The terms of art, "extreme hardship" and "urgent humanitarian reasons," refer to distinct processes in Immigration Law.
Extreme Hardship:
This is most often seen as a defense in Immigration Court, either in a cancellation of removal (42A or 42B) or in a waiver of forgiveness (Form I-601 or I-601A.) You are trying to overcome deportability based on the fact that your US citizen spouse, parent, or child would suffer extreme hardship in your absence.  You need to show extreme hardship in two contexts: the first is should the US citizen accompany the person being deported back to their home country, then there would be extreme hardship in perhaps education for their children, their earning potential, or maybe their personal safety. The second context extreme hardship needs to be proven is should the citizen not accompany their partner, parent, or child if they were deported then there may be extreme emotional hardship from being separated or hardship from loss of income. Showing extreme hardship should you be separated from your qualifying US citizen relative can clear the way for getting a green card or you may at least avoid deportation.
Urgent Humanitarian Reasons:
This is usually seen when someone is seeking an affirmative immigration benefit and time is a factor. In other words, the applicant is not in Immigration Court and is not using this as a defense. For example, the applicant may be seeking to obtain humanitarian parole which requires a compelling emergency and urgent humanitarian reasons to temporarily be allowed into the United States. These reasons could include a government coup in your home country, a financial crisis, or other human rights violations.