I agree with the age difference, but I also believe it could be their individual ages causing such a rebellion. Your 6 yr old is close to what I refer to as "the perfect age". At around 7, kids can be reasoned with, they know right from wrong and they have a strong desire to please people. At age 3, your other child is trying to find boundaries by pushing against every one, including her sister. She's also testing you to see how far she can go until you flip your lid, and let me guess...you try to get older sister to "make nice" because she is more reasonable, right? Add on top of that the fact they are both probably a little tired and grouchy when you all get home in the afternoons and you have the perfect ingredients for a meltdown, completely 100% at nobodies fault.
You need to set clear rules for them, no hitting, biting, etc. And I would suggest a small amt of time (20 mins or so) when you first get home for them to each go to a separate room, have a snack, enjoy some decompressing doing whatever relaxes each one, then you can allow them to engage with each other. If the battles begin in the car you need to set strict rules there, such as, if you are screaming and grabbing and hitting in the car, I will pull over and we will not move until you can control yourselves. If your 3 yr old LOVES a certain cd or insists on something being turned on for entertainment and the 6 yr old absolutely cannot stand it one more time, invest in headphones or ipod shuffles or something to just break up however the spat begins, but do not allow them to put yourselves or other cars in danger because you are trying to break up an arguement.
The good news is this, they WILL grow out of it! I remember when mine were about this age and they would argue over music in the car, the minute they started bickering I would flip it to Dr. Laura!! They knew right away they were stuck listening to a talk show and it was all their fault, cured it pretty fast :)
Good luck, believe it or not, they will be best friends one day!