Were there particular circumstances under which it was customary to vow the value of a person at all?
Not being acquainted with the popular customs of the Temple period, I couldn't say. I defer to others on answering this point.
[...] what governed whether you vowed the slave-valuation or the fixed valuation in Vayikra 27?
The determinant of whether you paid a fixed value erech or a neder nedavah of a slave valuation is solely the language used in the vow itself - if you vow someone's value, that's a nedava, while if you vow their erech, that's a fixed amount.
The Arachin makes some clear distinctions between these two options. For example, in the first perek (translation mine):
א,ב הנוכרי--רבי מאיר אומר, נערך, אבל לא מעריך; רבי יהודה אומר, מעריך, אבל לא נערך. וזה וזה מודים, שהן נודרין ונידרין.
The non-Jew -- Rabbi Meir says "They may be the subject of an erech-vow but may not take an erech-vow. Rabbi Yehudah says "They may take an erech-vow but not be the subject of an erech-vow. Both of these (Rabbis) agree that they may make a [donation] vow (of 'slave value') and be "vowed."
The question is whether the possukim in the Torah discussing the erech valuations is restricted to Jews (in that only Jews have a set erech) or whether the act of taking an erech-vow is restricted to Jews (but the erech evaluations discussed apply to all individuals). The former application holds that non-Jews may make the erech-vow to establish a monetary obligation (possibly an inclusion from ish), but it can only be made upon the value of a specified Jewish individual. The latter holds that only Israel, the audience in Vayikra 27, is capable to making the erech-vow but that language of the possukim allows for vowing of any human subject (possibly an inclusion from b'erkecha n'fashos).
Both opinions hold that any individual may establish a monetary obligation on themselves (or be the object of such an obligation) to the Temple by taking a neder - the possukim in Devarim (23:22-24) do not have any language that restricts the neder process to Jews (both as donators or as objects of the vow).
Another important distinction is made in the fifth perek (again, translation mine):
ה,ב דמי ידי עליי--שמין אותו, כמה הוא שווה ביד, וכמה הוא שווה בלא יד. זה חומר בנדרים מבערכים. חומר בערכים מבנדרים כיצד: האומר ערכי עליי, ומת--ייתנו הירושים; דמי עליי, ומת--לא ייתנו הירושים, שאין דמים למתים. ערך ידי וערך רגלי עליי, לא אמר כלום; ערך ראשי וערך כבדי עליי, נותן ערך כולו. זה הכלל--דבר שהנשמה תלויה בו, נותן ערך כולו.
[One who says] "The value of my hand is upon me [to donate]" -- we evaluate him, how much is he worth with a hand, and how much is he worth without a hand. This is the stringency of vows over arachim . How are arachim more stringent than vows? One who says "My erech is upon me" and dies -- their inheritors must give it. "My value is upon me" and dies -- the inheritors do not give it, since there is no value to the dead. [One who says] "The erech of my hand or foot is upon me" has said nothing. "The erech of my head or liver is upon me" donates their entire erech . This is the rule -- anything that the life is dependent upon, you give the entire erech.
The distinction made between a nedavah and an erech-vow is similar to that made between a nedavah and a neder vis a vis korbanos - the erech/neder is placed upon the oath-taker and therefore there is a lien on their property (the obligation isn't discharged by death). A nedavah is always on an object and the death/destruction of that object eliminates the nedavah (since, for example, the person or animal is now worthless).
Did they accomplish different spiritual or practical ends?
Hypothetically, based upon the end of 5:2 one can posit that the erech verses specify a valuation of the soul while the neder nedavah constitutes a de facto valuation of the body. Regardless, both erech and neder nedavah simply comprise financial "obligations" (and the method of discharging those obligations) voluntarily accepted by individuals to donate to the Bais Hamikdash. Any further distinction could be made in the minds of the donator.