WordPress 3.8.2 is a security update, so you likely won’t notice any changes to your site’s functionality. However, some of the security issues covered are fairly important, so make sure you update as soon as possible. You can check out the official WordPress release for update specifics and developer information.
Since there aren’t any major functionality changes, sites that were previously running WordPress 3.8 or 3.8.1 should work fine after an update. Just remember to back up your site before you press that button, just in case.