"Important" and "significant"

My American Heritage Dictionary shows the etymology of "important" as "[< Lat. importare, be significant]". So it's safe to say that in general usage, some meanings overlap. In certain technical or scientific contexts, however, the meaning of "signifcant" is more narrowly "having a meaning [at all]" (e.g., significant digit), whereas "important" is not a word with a precise meaning in such contexts.

In my lexicon, 'important' targets higher-level usage of the argument, while 'significant' is about lower level meaning:

"This is a significant modification, therefore it's inappropriate for this important release at the moment."

The importance of an event or an occasion is how worthwhile it is in the scheme of things.

The significance of an event or an occasion is the underlying meaning or interpretation of it.

Source: ENGLISH-TEST.NET

"Important" and "significant" are extremely close in meaning, but I think you are right that "important" is less formal than "significant". I haven't done the research, but I think we'd find that "important" is an older word in English than "significant", and that is the reason for the difference. If I had to bet, "important" entered the language around 1066, and "significant" came along during the renaissance.

Modern English was formed from the conquest of Germanic language speakers by Norman French speakers, with later overlays of educated people using Latin and Modern French words. Therefore the social pattern is that the newer the word, the higher its status; naturally this is mostly unconscious, but speakers attempting to establish high status will tend to use words of Romance origin.

Note that this trend it not limited to Germanic vs Romance words, or to specific historical periods. A modern example might be the status progression in "teacher/instructor/guru".