What is the difference between 'important' and similar words?

​While their meanings are relatively similar, their usages are different.

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Important vs significant

For example:

• It is important to avoid any exposure to the sun until your burns have healed.
• A significant demographic change was the decline of infant mortality at the turn of the century.​

The difference is that 'important' can be used with a reason for the thing being important (e.g. 'so your burns can heal'), while 'significant' is only a value judgement (i.e. the thing is important and should be paid attention to).
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Important vs essential

For example:

• This will be an important game for me to demonstrate my ability.
• A free press is an essential component of a democratic society.

The difference is a matter of degree. The game is more important than an ordinary game (so, 'important'), while a free press is really important (so, 'essential').
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Important vs crucial

For example:

• Have you ever had something important to say but you couldn't find the words?
• Management and cost control will be crucial in deciding success or failure.

The difference is that something 'important' can exist independently, while something 'crucial' has to play a role in determining an outcome.