Washington Manual is meatier, covers more topics. But its a little too big to fit well in the white coat (it'll fit, but thats all you'll get into that pocket).
Red book is golden, imho. Fits well in the white coat, and really has "the essentials" for all the bread & butter medicine stuff. If you know everything in the red book about whatever condition your patient has on rounds, you'll look great.
Don't really need both. If you need more info beyond rapid references like these, your best bet is to go to UpToDate or some other source.