How to copy only new files AND only those that are modified after a certain date?

This will probably get you what you want:

Robocopy c:\ e:\ *.* /MAXAGE:20130301 /XO /E

Add the /L command to this to see what it will do without it actually doing it first just to make sure.

Check here for more information on Robocopy's options.

Just use both options in the same command:

xcopy c:\*.* e:\*.* /D /D:03-01-2013

gets you what you want.