You pull a flap on airplane seatbelts, you push a button on car seatbelts. Why?
As Roe pointed out you don't necessarily pull a flap on airplane seatbelts -- Sometimes (particularly in newer light GA planes from the 1990s and on) you push a button just like you do in a car, other times it's the "utility buckle" (flap buckle) - sometimes with a shoulder harness attach peg or other attach mechanism, and sometimes you'll find rotary buckles, where you turn a knob to release the belt (particularly common in aerobatic aircraft, with 4-point or 5-point restraints). I've attached a little gallery of belt types below (and this isn't even complete - military aircraft hafe some downright funky buckles!).
This variety is why the FAA requires flight crew to instruct passengers in the operation of the seat belts.
So, why are the belts in airplanes different? And why are there so many types?
Each belt is good at something different, and each belt has its own set of drawbacks.
The automotive style push-button belt is very familiar to infrequent flyers.
Most people who are going to get on a plane have been in a car (they probably rode in one to the airport), so familiarity with automotive-style seatbelts can generally be assumed. This is great when you take someone for a flight who isn't a frequent airline passenger.
The drawback to the automotive belt is it's not as easy for rescuers to operate, and the push-button mechanism can jam - particularly in fires where the plastic shell can melt/fuse (in cars the belts are often cut by rescuers because it's easier/faster than dealing with the button). If your passengers are injured in an emergency and can't operate the belt it may be difficult for you to locate and press the button to assist in evacuation.
The "utility buckle" flap belt is easy for rescuers & cabin crew to operate.
The exact opposite of the push-button belt's chief drawback: It's really easy for cabin crew or rescurers to find a utility buckle on a belt by feel and yank the flap to release it (try it some time: Buckle someone into an airplane seat, close your eyes, and find/release the buckle. The hardest part is finding the belt if they're wearing bulky clothes). It's also relatively hard to jam a utility buckle, and relatively easy to inspect them to determine if they're damaged.
The drawback to the utility buckle is that not everyone has been on an airplane (or in a really old car that uses them), so they might not be familiar with "Lift the flap to release the belt", and in a panic situation it may take them a few seconds to remember what to do.
The rotary buckle is unique and kind of awesome for what it's built for.
Rotary buckles make it easy to attach 4 or 5 point restraints, and to release all or part of them quickly. (The buckle shown in the photos is single-function - turn to release - but dual-function buckles exist where you can press the button in the center of the buckle to release the shoulder straps, or turn the knob to release everything.)
The big drawbacks to rotary buckles are their novelty: On the ground you usually only find them on 5-point racing harnesses, and in the air they're pretty uncommon too (mainly aerobatic aircraft as I mentioned earlier). They're probably the last type of buckle a passenger or rescuer would expect to have to deal with, and particularly with the dual-action buttons they can present a challenge in a panic egress (passengers will hit the button and release the shoulder harnesses, but they're still restrained by the lower portion of the belt - this can cost precious time for them to realize they have to turn the knob).
I think the classic economy class buckle is just a compromise: On cars, you can easily reach the side of the seat, and it's more comfortable to have it on the side than a block of metal in the middle. In aircraft seats however, it's difficult to reach down in between, and if something happens quickly, you want to easily release it. The design itself might have a historical element, since a seatbelt which is very easy to open might make passengers feel safer, especially when many were new to flying.
That being said, there are several business-class seats today which utilize conventional seatbelt styles.