When register my site to Google webmaster tools, which one should I register?
If you are serving the same content under both domain names and you do not have the option to redirect (301/permanent) visitors who land at the non-canonical domain over to your canonical domain, you should register both domains at Webmaster Tools and then set your preferred domain.
I registered mine with www. I suppose there isn't much of a difference between them. Using www.yourdomain.com or yourdomain.com is just a different referral of the A-record, but resolves the same host IP.
You must register both. It will help you to verify your preferred domain and also will help to monitor both domains.