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"Important" and "significant" seems to be very close in meaning when denoting that something matters much.

But am I right in thinking that "important" is less formal word than "significant"? And what, if any, are other differences in usage between them?

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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.

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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

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"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".

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The OED’s earliest citation for ‘important’ is dated 1586. That for ‘significant’ is 1566. –  Barrie England Oct 6 '11 at 6:32
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Significant also has implications of relative or quantifiable importance, or importance within a system. "Significant" is a slightly more exacting and fussy version of "important", analogous to the difference between "exact" and "precise".

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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."

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Both words have more than one definition and are used differently. There is rather more to it than one being formal and the other not. There may be some contexts in which they are interchangeable, but there are many in which they are not. There are long entries on both in the OED and it would be extremely ambitious to try to summarise the various definitions and uses in a few lines. As just one example, statisticians will typically speak of a number or a percentage as being significant when they want to say something about its precision. To say, on the other hand, that a number is important suggests that it has a value that extends beyond the immediate mathematical environment.

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