Should the website contain information which is discouraged to have in the CV? Here and there in this site, I get that you should have your own website. It is one way to networking, making you standing out of the crowd. It can also be your extended version of CV or SOP, helping others knowing you more.

But what about the information that is discouraged to have in such documents? For example:


Contests. A math contest indicates that you can solve hard problems, but it doesn't say anything about your research ability. A chess contest is worse, it doesn't even relate to your field.
Courses.
Volunteer activities.


The main point is that they show that you are not investing your time properly. But does this contrast to the "a place to show yourself more" point?

And how about putting things like life objection, music I like, drawing my kid draws, etc? These are obviously not suitable for a CV, but they are just hobbies. Everyone knows that investing your time on this isn't a bad thing for your career, but a way to balance your life.



Links:


Is web-presence important for researchers?
Are participations in competitions well recognized for a Ph.D. application?  
Can I include the completion of Udacity and Coursera classes I have attended in an academic CV?
Why are contributions to the education of the broader public not valued much within academia?
In moderation, I don't see anything wrong with including content on your website that you would not include on a CV.  In principle it could hurt your career if you post something ridiculous or objectionable, but information about chess, ordinary volunteer activities, children, etc. doesn't seem problematic.  The main issue is making sure it doesn't overshadow the academic content: if you provide more details about your chess activities than about research, then it may give the impression that you are devoting more time and attention to chess than to research.  (Even if that's not the case.  For example, a grad student who is just starting with research but has a lengthy chess history might give the wrong impression, regardless of current priorities.)  If you want to cover non-academic activities in substantial detail, then it may be safer to set up a personal website that is clearly separated from your academic site.