64 bit programs have to be better than 32 bit programs, right? This is a question that keeps popping up in my classes and the various support forums, usually in the form of "xxxxx does not work in SharePoint". In general all client applications used with SharePoint should be 32 bit. This includes Word, Excel, IE, etc. The Office ActiveX controls used with SharePoint are all 32 bit.
Here's the SharePoint 2010 Browser Support article from TechNet:
For specific things not supported in 64 bit see:
Things have gotten better with SharePoint2013 as there's less dependence on ActiveX controls. The Office 64 bit article (https://technet.microsoft.com/en-us/library/ee681792.aspx) says "We recommend the 32-bit version of Office for most users, because it's more compatible with most other applications, especially third-party add-ins."
For SharePoint 2013 the support browsers article says it a little differently, but has the same message: "Some functionality in SharePoint 2013 requires ActiveX controls. This produces limitations on browsers which do not support ActiveX. Currently only 32-bit versions of Internet Explorer support this functionality."