Why are passwords so important?

Because they are the first line of defence against interactive attacks on your system. It can be stated simply: if a cracker cannot interact with your system(s), and he has no access to read or write the information contained in the password file, then he has almost no avenues of attack left open to break your system.

This is also why, if a cracker can at least read your password file (and if you are on a vanilla modern Unix, you should assume this) it is so important that he is not able to break any of the passwords contained therein. If he can, then it is also fair to assume that he can (a) log on to your system and can then (b) break into "root" via an operating system hole.