We encourage people to make them the same on both sites if they have an account on the old site. That's just for the sake of simplicity when you log in.

There is also bug in IE9 which further complicates it for people using IE.  We are working on resolving this and it's discussed in more detail here. 

Update:
If you change your password on the new site it will update your password on the old site. This will allow you to keep you password the same to avoid confusion. We do NOT plan on doing the same on the old site. If you change you password on the old site it is only changed on the old site. 

Feel free to give us feed back on this update in the comments below. 
Nick