You are now in the territory of subtle differences, and if you really want to know, your best strategy is to look at a good dictionary. I personally like the American Heritage dictionary. I own a paper copy, but it is online at ahdictionary.org.
Dictionaries go beyond basic meanings; good dictionaries show you range of meanings and nuances.
I really like their usage notes, when they have them.
Whenever I bother to consult a dictionary, I ALWAYS learn something interesting.
I'm not trying to be unhelpful, I'm serious. When you get to the level of distinctions like "significant" versus "important" it's time to go to the dictionary.
As a general rule, when in doubt use the shorter, simpler, more common word, and express your meaning with the rest of the sentence.