We drive about 18 hours too. I've done both. Our kids are excellent travelers. I highly, highly
HIGHLY
recommend stopping. Did I say STOP!! yet?
Ahem. Well.
We start at 4 am. The kids sleep the night before in cozy clothes, and trundle sleepily into the van with their blankies and pillows.
Around 8 am we stop for doughnuts and coffee.
Around 12 noon we get go to a grocery store and buy some sandwich stuff. We play hide and seek and jump rope at a handy park or rest stop.
Around 3 pm we start to stop at the welcome centers, picking up the hotel flyers to see who has a pool, wi-fi and a decent price. Complimentary bfast is a plus too.
DH decides to stop around 5 to 6 pm. We swim. We eat. We chortle madly about Disney. We sleep. We get up about 7 am. Have breakfast, maybe swim again, and head out.
The advantages of this system is that DH doesn't get into a driving obsession and feel the need to keep going. We're allowed out every couple hours to pee. Our knees don't feel like they'll never straighten again. No one threatens to kill the children within their earshot. We're all still feeling happy about Disney and going together the next day. No one cries.
However, my experience with traveling straight through...
Kids are too excited to sleep much on the way down.
Kids are so excited they are bored and bouncy, shrill and shreeky.
Kids are given snacks and drinks to shut them up.
Kids have to pee.
Driver becomes exasperated with amount of stopping.
Driver needs to stop in unfamiliar places in the middle of the night for peeing.
Driver gets on wrong road, or the right road going the wrong way. Or the right road going the right way, but it's so blasted DARK in North Carolina at night, who KNOWS where you are anyway!!! Ahem.
So, after escaping the military school grounds, you get back on the interstate. Only to keep going. And going. And going.
Children begin to moan uncontrollably with despair. Oh wait, that was me.
By the time you get to Disney, you are fried. Beyond exhausted. I was awake the one time we did it for over 38 hours. Our shirts all read "We're too excited to sleep!" I was. I was not a nice woman, however.
We drove straight through with two drivers, two elementary school kids, and 9 high school seniors. And me. I was shotgun the whole time. I kept the drivers awake. I plugged the various ipods in to the cassette adaptor, navigated, and interpreted the various shrieks and moans into intelligible commands for the driver, "I need to pee." "I need to get out NOW." "If we don't stop right NOW I'm going to puke." "Even if we do stop NOW, I'll puke, so stop NOW so I can do it outside."
Did I mention that my daughter and son developed strep while in Disney on that trip? On the way home I sat behind the drivers, on a short bench seat. My knees never unbent. Everytime I started to drift off, my head would sag, causing my neck to spasm and snap me back to wakefulness. DS just leaned on me and was sad. DD puked in every state on the way home except Maryland and Pennsylvania. In the middle of the night. In the dark on I95. Stumbling onto the shoulder of the highway in the dark. Puking helplessly. Couple times she tumbled down the side of the road into the ditch.
Only once did I step on a fire ant hill in my sandals while trying to help her.
DH offered to stop for the night with only 4 hours to go, but we were all so numb with despair, illness, and exhaustion that we just stared at him in horror. Prolong this agony? PLEASE just get us HOME!
I do not recommend driving straight through with children.
Trust me when I say that if you decide to do so anyway, I'll pray for you.
. That was the best warning against an all night drive!
On the way home from WDW last time we stopped for dinner in Savannah and I know we left Disney late that day because we had lunch on the beach in Daytona (only an hour or so from Orlando). I remember thinking it was an easy drive to Savannah from WDW and we took our time drove small HWYs, went sightseeing etc.... Have fun fun fun!
I think every 5 minutes I change my mind. 
