You do get way less experience for killing things with the vehicle vs. killing them while on foot. I'm honestly not sure about the quality of drops being different. But in my experience playing, if I cared about xp at all, I always stepped out of my car. I don't remember really specific numbers off the top of my head, but for example, killing a Badass Spiderant King with the vehicle awareded me half (less even, I believe) the xp versus stepping out and shooting it. This is not a bug and is working as intended, so you can't just cheese your way through the game.
Later on in the game, enemies will tend to do too much damage to your vehicle too quickly for it to be useful for seriously clearing out areas. If you're just going from point A to point B, sure, you can often run over whatever you want on the way (keeping an eye on vehicle health of course). I would park it outside of a hideout though if you're going in, as enemies will chew through it fairly quickly later on and it'll explode, leaving you pretty damaged, if not already on your knees needing a Second Wind.
Also, when you start your second playthrough, the enemies damage the vehicle way more when you run into them and will sometimes explode after hitting one enemy!