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Without going into the fact the Cryptography has been used in some form or another for many hundreds, if not thousands of years, it's benefits when it comes to computers really come into their own. I will give some examples.
Cryptography allows us to hide information. Be it a letter, be it a file, be it a constant stream of video or audio over digital sources. Now computers are good at only one thing it's manipulating and processing truly staggering amounts of data and information of many kinds in a short period of time. Information that would take humans to work though many, many times longer.
There for the basis of breaking encryption becomes much easier with computers and logical/Turing machines which are constructed to apply mathematical force to a cytological methods until it finds the key. Normally it would take a bunch of people working on paper and pencil a insanely long time due to the complexity of modern cryto-methodology. So much so it's just not remotely feasible without some kind of computer. In fact, computer technology as we know it today is largely based on the work of men who created them as tools to, in fact, try and break codes during World War II.
So since computers make breaking codes much easier it therefore becomes necessary to use stronger and stronger methods of encryption and increasingly more complex keys. This has been a constant war for some time between those who built crypto-systems and those who break codes.
Since most of the data in the world these days flows over computers on either public or private network connection ranging from local to global it becomes necessary to have a strong, reliable encryption methods for your data because it's laughably easy for some one to just sniff packets off a network and read all the data you happen to be sending if it's not encrypted. So Crypto-systems are needed in order to maintain data secrecy (for Governments, Companies protecting intellectual property, individuals who seek private communication and yes, even criminals who seek to keep their data safe from the authorities) for the people in question.
The benefits could be compared to either pulling some one aside to speak to them privately (encrypted) or shouting the information across a crowded room (unencrypted). It all depends on your need for security and privacy.
This is a complex subject, but basically encrypting data if done correctly allows me to hide what I am saying to my friend/co-worker/boss from those I might not want to read my mail/conversations/files.
Hope that helps!
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