What is the difference between http://example.com and http://www.example.com.

Both 'http://example.com' and 'http://www.example.com' are technically different but practically the same. This means that though the search engines reads the two as different URLs the result you see will be the same most of the time.

If they are not much different should you be bothered about it?

Yeah, a bit. Because the search engine reads the two URLs as different, every data in the search engine will be different for the two. Data I mean hear is about the link popularity, search keyword, cache, etc. The problem here is that if half the people are coming to your site by non-www link and half by www link your popularity also will be halved. You might see both of them as in a result for a search with two different cache if the cache were taken at different times and you have updated your site in between.

What you should do in this case?

I suggest you do a permanent redirect to your preferred URL. This will ensure that people coming through both URL will come to same place. The Permanent redirect is otherwise called 301 redirect. You can read how to create a permanent 301 redirect here. This is the way of telling the search engine that you wish to show your preferred URL and the process is known as canonicalization of URL. The 'to redirected' URL is known as canonical URL.