They are different in both function and form.
Here's my take:
All emergency brakes are parking brakes, but not all parking brakes are emergency brakes.
The lever between the seats is often called an "emergency brake", as in the event of a brake failure, you can hold the button pressed and vary the amount of pull on the lever to come to a controlled stop. Thanks to the ratcheting mechanism, it doubles as a parking brake when not holding the button.
The foot pedal variety in SOME cars works the same way, as you can keep the release lever pulled and vary the pressure with your left foot to control your stopping. It's not ideal, as you have to dive under the dash to reach the release, but it's better than crashing.
But SOME cars have a push-on push-off style of pedal here, which really makes it a parking brake "only", and not a good emergency brake. With no practical way to modulate the braking force, and no way to release without fully applying, you could find your vehicle spinning out of control after locking up two of your wheels. Better than crashing from having no brakes, but not a good situation.
Some newer cars have the "electronic" parking brake, where a single button press applies or releases the parking brake. Basically a system designed around the fact that modern dual-circuit hydraulic primary brakes are reliable enough, that using that center space for cup holders is an acceptable tradeoff.