As something of a fossilized and archaic term, 何卒{なにとぞ} comes across as more formal and stiff. This would not be used in everyday talk. Possible use cases in the closing of formal correspondence might be:
- ご質問{しつもん}等{とう}がございましたら 何卒 ご連絡{れんらく}ください。
- 何卒 宜{よろ}しくお願{ねが}い致{いた}します。
- 何卒 どうぞ宜{よろ}しくお願{ねが}い致{いた}します。
Some folks consider this last one to be redundant or incorrect with both 何卒 and どうぞ, as these have basically the same meaning, but I do find numerous examples of use by what appear to be native speakers, so take that for what you will.
Breaking this term down, Shogakukan explains that this is 何{なに} "what" + と (particle) + ぞ (particle), with the 卒 character serving as ateji. The ぞ isn't used much and might be less well-known. In this context, it basically serves as an intensifier, a bit like も after question words. The resulting meaning of the phrase is a bit like English "whatever the case may be", or "no matter what". Synonyms listed in the dictionary entry are どうぞして, なんとかして, and どうか. (Incidentally, the ぞ in なにとぞ is the same ぞ in どうぞ.)