What is effect of linking with Non-Similar URL?
The Major impact is decrease in link popularity as search engines consider them as two URL's and count link popularity separately for both.
 
How to verify that my site both version exit in search engines.
You can just the Google PR of your pages with www version and non www version. You will find different PR for both versions
 
What to do to prevent this.
Select same form of your URL for inbound and internal linking. Make sure all your incoming links are pointed to the same form of your URL. When setting up your links make sure to stay consistent with your URL structure. Follow the same structure within your own website. Make sure that your own internal links on your website link to the same URL as your inbound links.
 
Which Format is to be adopted?
 
There's absolutely no difference for Search engines for using www or non-www version. Different people have different views Some say with www Version you have to type four more character. Where as lot have opinion that all user's type first www while opening on browser.Many directories require the WWW in the site location so it is preferable to use the WWW version if you can www is very common, people can get confused if you have a domain without. Don't forget, in your browser when you type "SEO" in the address bar and hit ctrl+enter, it automatically adds the www along with the rest of the URL. For that reason alone, some suggests www version. You may adopt any version of URL but consistency should be there in internal linking.
 
What to do If both URL structure exist on old site!
 
If you have both yoursite.com and www.yoursite.com you must consolidate two "sites" into one.
Showing your content under two different URLs is never a good idea.

First find out the PageRank of each version. and can choose version which has higher PR as it is mostly linked by other websites. You must adopt 301 redirect so there aren't two versions of your site out there (www.yoursite.com and yoursite.com). It will most likely change your rankings positively as you no longer "split" your site across two sets of URLs, hence each URL will get more link power. You may not see the effect in short term as search engines takes time in whole update of web. You can make 301 redirection in many but most popularity is to use either .htaccess file or set on server. You can also set up your server with the right default hostname when you do this stuff (or to have your host do it for you).If the web server is not configured to have the proper default (www or non-www), there will be so many redirects, it will take ages to load the site.
 
By now providing a redirect, people will naturally link to you using the one that you want to be used. Make sure that all of your internal links go to the correct one too.
 
Linking of your site with consistence URL will force the search engine to index real URL and people to link the URL structure you want and will result into saving of leakage of PageRank and increase in link popularity to improve you search engine ranking and goal of each web site owner to increase traffic will be achieved.