Why do the F-22 and F-35 look similar to each other and different from previous designs?
ymb1 is right - stealth and not styling determine the looks of both the F-22 and the F-35. But there is more to it.

Watch Lockheed test pilot Tom Morgenfeld talk about them in this video:


                
            

The juicy bits are at 37:15' and at 43:35' into the video. Concerning the F-22 he says: Northrop built the F-23 exactly to what the Air Force asked for, while Lockheed built what the Air Force secretly wanted. And that was another F-15, with similar manoeuvrability.

Concerning the F-35 he says Lockheed's goal was to build an aircraft which is at least as good as an F-18 around the ship. This means manoeuvrability to land on a carrier even from off-nominal conditions; very forgiving to fly. By using the F-22 baseline, this manoeuvrability could be ensured.

Now look at all three of them:


McDonnell-Douglas F-15 Eagle (picture source)


Lockheed-Martin F-22 Raptor (picture source)


Lockheed-Martin F-35 Lightning II (picture source)

They look similar, don't they? Differences like a forward swept trailing edge and canted verticals are due to stealth, as is edge alignment.