Any time that DNS changes are made (on any level), you need to wait for propagation to complete. Propagation usually takes from several hours to 5 days, but can take as long as 7 to 10 days. During propagation, traffic may come to either location. One person may see the new server while someone else sees the old one. Also, yourdomain.name may work while www.yourdomain.name does not. All of this is normal during propagation.
Note that this means that just because you go to the new site when you type in the domain name does not mean that propagation is complete. The best way to determine whether or not propagation is complete is to review the statistics for each site (both the old one and the new one).