As alluded to in Calchas's answer, the usefulness of a UN laissez-passer (LP) depends entirely on the country to which its holder is traveling. From (indirect) personal experience, I can report that practice varies widely.
In the United States, at least, diplomatic and semi-diplomatic privileges and immunities are granted based on a person's accreditation to the Department of State and on their rank and function in the UN system, not on the possession of any particular document. See, for example, 22 USC 288d, Privileges, exemptions, and immunities of officers, employees, and their families; waiver.
As a consequence, for example, someone who holds a UNLP but travels to the US for personal reasons using a B visa or the visa waiver program derives no privileges or immunities from the LP. Other countries, of course, may choose to grant privileges or immunities to similarly situated people.
As noted by Zach Lipton in a comment, the LP may be useful for UN employees traveling to a country that would not otherwise grant admission because of the employee's nationality. This does not apply in the US, however, because the US has developed other mechanisms to deal with this issue, among which is the C-2 visa.
Is it for proving UN affiliation within a country?
It is certainly one way to do that, although UN employees generally also have an employee identification card.
Without any particular diplomatic privileges is this useful outside of war zones, refugee camps, uranium-enrichment plants, etc?
In many countries, the document does come with diplomatic privileges, but even in those in which it does not, it does serve in some cases to qualify the bearer for a visa exemption. To restate its purpose generally, it serves as a travel document that identifies the bearer as a UN officer or employee. What any country chooses to do with that information is up to that country.