Gravity can be described as a pure acceleration, and it is perceived at such. You'd feel 0G when in free fall and when flying at constant velocity in outer space, and you'd feel 1G when standing still on the ground. If, in addition, you accelerate forward in a car with 1G, your total acceleration would amount to ca. 1.4G.
However, whereas your body senses all sources of acceleration in the same way, it is not as well accustomed to non-gravitational accelerations, its non-zero derivatives and certain related sensor inputs, as it is to gravity. The change of feeling moreover intensifies when sensor signals do not meet with your expectations or each other, or when they have high derivatives.
As for your premise that the difference in perception of acceleration is caused by the fact that gravity applies to your body as a whole whereas the acceleration in a car is the result of a contact force, it is not correct. The force balances are similar. In both cases the contact force is in equilibrium with an inertial force as the result of an acceleration of every point in your body. With gravity, the contact surface (when standing) is the bottom of your feet, on which a force is exerted equal to that required for accelerating every point in your body up with 1G. When accelerating in a car, the contact surface is mainly your back, on which a force is exerted equal to that required for accelerating every point in your body forward with the actual acceleration. The problem in your thinking might be related to a swap of cause and effect (gravity causes the reaction force; car acceleration is caused by the reaction force), even though Newtonian physics don't actually care about this. After all, force balances are about instantaneous equilibria: every action is also a reaction, and vice versa.
I'll break the main answer down in several points for the interested reader.
The human body does in fact sense acceleration and thus gravity directly, in contrast to what most answers here claim.
I am surprised to read all the claims stating that humans cannot sense acceleration directly. This is incorrect. The vestibular system, located behind your ears, allow you to directly sense accelerations. More specifically, it consists of the otolith and the semicircular cannels, which allow you to sense linear and rotational accelerations respectively. They help you balance, and they are responsible for making you feel nauseous after jerky rollercoaster rides. If you want to know how it feels like to have a malfunctioning vestibular system: get drunk*. And no, the vestibular system does not distinguish gravity from acceleration, for the fact that gravity is an acceleration. By combining the gravitational acceleration and that of a car, the amplitude of the acceleration vector increases and its orientation changes.
For your interest:
There is another important sense in the human body of which many are not aware, namely proprioception. It senses internal body kinematics and kinetics, using the golgi tendon organs in your muscles. They close the loop with muscle control. The role of proprioception in daily activities such as walking is very important and shouldn't be confused with touch. Watch this short 2 minute video to understand how it's like to go around without proprioception.
The human body is a master of sensor fusion, but it can be fooled.
Your central nervous system fuses many types of sensor signals and your anticipation/expectation/knowledge for purposes of understanding and controlling all sorts of activities. These sensor signals include touch, vision and the aforementioned acceleration and proprioception. When you hit the accelerator in your car, you'll feel relatively comfortable. Everything matches up: the acceleration, the compression in your seat, the vision and your own anticipation. Instead, when you are a co-driver, it will easier feel less comfortable, because you lack the anticipation.
For your interest:
There are also situations where sensor signals do not match, which cause confusion and discomfort. Consider for example being in a parked train, and the train(s) next to you start driving. Your vision might make you think that you start moving, but your vestibular, touch and proprioceptive senses disagree, causing you to feel discomfort up to the point that you realise that your sensor fusion made a mistake. These illusions of self-motion are well known and called vection. The initial mistake occurs because your sense of vision is typically given more value. That said, your sensor fusion system can actually filter a non-matching sensor signal quite well. This has allowed for inventions such as flight simulators to feel realistic. The vision, sound, surrounding (realistic cockpit) and seat compression as caused by slightly tilting the platform all make you think that you are flying a plane. The only missing aspect is the actual linear acceleration (although not entirely, because the platform can move back and forth several meters, and mimic vibrations such as those caused by turbulence).
Over the long haul, the human body is a master of adaptation.
So why is it then, that we feel the effects of accelerating in a car more prominently (even if we hit the accelerator ourselves) than, say, walking? Let's put this to the extreme, why do we feel discomfort or excited, when jumping out of an aircraft? After all, the inputs match up: we have vision, we feel the acceleration (or actually, the lack thereof when jumping from the plane), we feel the air drag and we hear it too. Last but not least, we anticipated it (it was your own choice, nobody pushed you). The answer is simply that we are not used to it. Other than a counter-intuitive mental state for the case of jumping (our instinct didn't quite teach us that it would be a good idea to jump several kilometers down), our bodies are physically adapted to the presence of a ca. 10m/s^2 downward acceleration: our gravity. You can thank evolution for this. For the same reason, we have no problem being around in a pressurized environment of ca. 100MPa: our atmospheric pressure (rather the contrary, you'll not feel at comfort at all if you take that pressure away...). It exerts a force similar to the Earthly weight of ca. 10 Renault Twingos per square meter of surface, but our body does too in the opposite direction, and our touch sensors are trained to ignore their mutual presence. Same for acceleration: not only does the orientation change, but also the amplitude (0 for a free fall and >1 when accelerating a car). Worse than the difference, is the change: the vestibular organ is not particularly keen on change of acceleration, known as jerk.
For your interest:
That said, whereas the human body has gone through a long optimization process as caused by evolution, it is still very well capable of adapting over a much shorter time interval than that of a life time, within a certain window of capabilities. Astronauts are capable of adapting to a 0G environment after a while (although it is not healthy for them), and you might be familiar with the human's capability to develop sea legs when sailing on a boat for long enough, etc.
* I cannot be held responsible for any effect this experiment might have.