I think what's really going on here can be traced back to the two different ways 形容動詞 (けいようどうし: adjectival nouns or "な-adjectives") were inflected. If we look under the 連用形 (れんようけい: the "adverbial inflection", for lack of a better term) column under the first table on this Wikibooks page detailing Classical Japanese inflection patterns, we find the following two patterns:
ナリ活用 (the ナリ inflection), such as with 静か【しずか】: 静かなり or 静かに
タリ活用 (the タリ inflection), such as with 堂々【どうどう】: 堂々たり or 堂々と
(The Wikipedia page on 形容動詞 mentions that these inflections were derived from the ～にあり and ～とあり forms, respectively, in Classical Japanese.) For the most part the old styles of inflection have fallen out of use, but there are many examples fossilized in Modern Japanese:
聖なる夜 holy night (also used as the Japanese title of "Silent Night")
暗澹【あんたん】たる時期 a dark period
So without getting into the entire class of words that is used as adverbs without any special と or に appendages, we can see that from early on 形容動詞 were split in that some took なり／に to become adverbs, while some took たり／と.
- If a word is more commonly used as a な-adjective (that is, if its 連体形 (れんたいけい), the form by which it connects to a noun, is な), the adverbial form will have に.
- If a word typically does not connect to nouns or is more commonly used as an adverb, the adverbial form will usually have と or nothing at all (excepting common forms such as ～になる or ～にする).
These are general rules (and not very good rules at that), and so exceptions, such as 次々, aren't hard to find.
静か｛○に／×と／×∅｝話す speak quietly
きれい｛○に／×と／×∅｝整える arrange neatly
ゆっくり｛△に／○と／○∅｝歩く walk slowly
きっぱり｛△に／○と／○∅｝言う say flatly
はっきり｛△に／○と／○∅｝見える [be able to] see clearly
Unfortunately there are no pretty dividing lines between に and と here. This is probably because に and と have both retained their role of "adverbializer" throughout the evolution of Japanese, and neither form succeeded in displacing the other. Boaz Yaniv mentions this phenomenon in his answer for why some adjectives use な and some use の.